Given sin-1 (2a/ 1+ a2) + sin-1 (2b/ 1+ b2) = 2 tan-1 x
$\text { If } \sin ^{-1}\left(2 a / 1+a^{2}\right)+\sin ^{-1}\left(2 b / 1+b^{2}\right)=2 \tan ^{-1} x, \text { prove that } x=(a+b / 1-a b)$
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Maths
Prove that: (i) $\left.\left.\tan ^{-1}\left\{\left(1-x^{2}\right) / 2 x\right)\right\}+\cot ^{-1}\left\{\left(1-x^{2}\right) / 2 x\right)\right\}=\pi / 2$ (ii) $\left.\sin \left\{\tan ^{-1}\left(1-x^{2}\right) / 2 x\right)+\cos ^{-1}\left(1-x^{2}\right) /\left(1+x^{2}\right)\right\}=1$
$(i)$ $(ii)$
$\text { If } \sin ^{-1}\left(2 a / 1+a^{2}\right)-\cos ^{-1}\left(1-b^{2} / 1+b^{2}\right)=\tan ^{-1}\left(2 x / 1-x^{2}\right), \text { then prove that } x=(a-b) /(1+a b)$
Prove the following results: (ix) $2 \tan ^{-1}(1 / 2)+\tan ^{-1}(1 / 7)=\tan ^{-1}(31 / 17)$ (x) $4 \tan ^{-1}(1 / 5)-\tan ^{-1}(1 / 239)=\pi / 4$
$(ix)$ $(x)$
Prove the following results: (vii) $2 \tan ^{-1}(1 / 5)+\tan ^{-1}(1 / 8)=\tan ^{-1}(4 / 7)$ (viii) $2 \tan ^{-1}(3 / 4)-\tan ^{-1}(17 / 31)=\pi / 4$
$(vii)$ $(viii)$
Prove the following results: (v) $\sin ^{-1}(4 / 5)+2 \tan ^{-1}(1 / 3)=\pi / 2$ (vi) $2 \sin ^{-1}(3 / 5)-\tan ^{-1}(17 / 31)=\pi / 4$
$(v)$ $(vi)$
Prove the following results: (iii) $\tan ^{-1}(2 / 3)=1 / 2 \tan ^{-1}(12 / 5)$ (iv) $\tan ^{-1}(1 / 7)+2 \tan ^{-1}(1 / 3)=\pi / 4$
$(iii)$ $(iv)$
Prove the following results: (i) $2 \sin ^{-1}(3 / 5)=\tan ^{-1}(24 / 7)$ (ii) $\tan ^{-1} 1 / 4+\tan ^{-1}(2 / 9)=1 / 2 \cos ^{-1}(3 / 5)=1 / 2 \sin ^{-1}(4 / 5)$
$(i)$ $(ii)$
Evaluate the following: (iii) $\operatorname{Sin}\left\{1 / 2 \cos ^{-1}(4 / 5)\right\}$ (iv) $\operatorname{Sin}\left(2 \tan ^{-1} 2 / 3\right)+\cos \left(\tan ^{-1} \sqrt{3}\right)$
$(iii)$ $(iv)$
Evaluate the following: (i) $\tan \left\{2 \tan ^{-1}(1 / 5)-\pi / 4\right\}$ (ii) $\operatorname{Tan}\left\{1 / 2 \sin ^{-1}(3 / 4)\right\}$
$(i)$ $(ii)$
$\text { Solve the equation: } \cos ^{-1}(a / x)-\cos ^{-1}(b / x)=\cos ^{-1}(1 / b)-\cos ^{-1}(1 / a)$
Since, cos-1 (a/x) – cos-1 (b/x) = cos-1 (1/b) – cos-1 (1/a)
$\text { If } \cos ^{-1}(x / 2)+\cos ^{-1}(y / 3)=a, \text { then prove that } 9 x^{2}-12 x y \cos a+4 y^{2}=36 \sin ^{2} a$
$\text { Evaluate: } \text { Cos }\left(\sin ^{-1} 3 / 5+\sin ^{-1} 5 / 13\right)$
Given Cos (sin -1 3/5 + sin-1 5/13)
$\text { Find the value of } \tan ^{-1}(x / y)-\tan ^{-1}\{(x-y) /(x+y)\}$
Prove the following results: (iii) $\tan ^{-1}(1 / 4)+\tan ^{-1}(2 / 9)=\operatorname{Sin}^{-1}(1 / \sqrt{5})$
$(iii)$
Prove the following results: (i) $\operatorname{tan}^{-1}(1 / 7)+\tan ^{-1}(1 / 13)=\tan ^{-1}(2 / 9)$ (ii) $\operatorname{sin}^{-1}(12 / 13)+\cos ^{-1}(4 / 5)+\tan ^{-1}(63 / 16)=\pi$
$(i)$ $(ii)$ LHS
If $\left(\sin ^{-1} x\right)^{2}+\left(\cos ^{-1} x\right)^{2}=17 \pi^{2} / 36$, find $x$
Since, $\cos ^{-1} x+\sin ^{-1} x=\pi / 2$ => $\cos ^{-1} x=\pi / 2-\sin ^{-1} x$ Substituting this in $\left(\sin ^{-1} x\right)^{2}+\left(\cos ^{-1} x\right)^{2}=17 \pi^{2} / 36$ $\left(\sin...
$\text { If } \cot \left(\cos ^{-1} 3 / 5+\sin ^{-1} x\right)=0, \text { find the value of } x$
$\cot \left(\cos ^{-1} 3 / 5+\sin ^{-1} x\right)=0$ => $\begin{array}{l} \left(\cos ^{-1} 3 / 5+\sin ^{-1} x\right)=\cot ^{-1}(0) \\ \left(\operatorname{Cos}^{-1} 3 / 5+\sin ^{-1} x\right)=\pi /...
$\text { If } \sin ^{-1} x+\sin ^{-1} y=\pi / 3 \text { and } \cos ^{-1} x-\cos ^{-1} y=\pi / 6, \text { find the values of } x \text { and } y$
Given sin-1 x + sin-1 y = π/3 ……. (i) And cos-1 x – cos-1 y = π/6 ……… (ii)
$\text { If } \cos ^{-1} x+\cos ^{-1} y=\pi / 4, \text { find the value of } \sin ^{-1} x+\sin ^{-1} y$
Since, cos-1 x + cos-1 y = π/4
Evaluate: (v) $\cos \left(\sec ^{-1} x+\operatorname{cosec}^{-1} x\right),|x| \geq 1$
$(v)$ $=>0$
Evaluate: (iii) $\operatorname{Sin}\left(\tan ^{-1} x+\tan ^{-1} 1 / x\right)$ for $x>0$ (iv) Cot $\left(\tan ^{-1} \mathrm{a}+\cot ^{-1} \mathrm{a}\right)$
$(iii)$ $(iv)$
Evaluate: (i) $\operatorname{Cot}\left(\sin ^{-1}(3 / 4)+\sec ^{-1}(4 / 3)\right)$ (ii) $\operatorname{Sin}\left(\tan ^{-1} x+\tan ^{-1} 1 / x\right)$ for $x<0$
$(i)$ $(ii)$
Evaluate: (iii) cot $\left\{\sec ^{-1}(-13 / 5)\right\}$
$(iii)$
Evaluate: (i) $\operatorname{Cos}\left\{\sin ^{-1}(-7 / 25)\right\}$ (ii) Sec $\left\{\cot ^{-1}(-5 / 12)\right\}$
$(i)$ $(ii)$
Evaluate each of the following: (ix) $\operatorname{Cos}\left(\tan ^{-1} 24 / 7\right)$
$(ix)$ .
Evaluate each of the following: (vii) Tan $\left(\cos ^{-1} 8 / 17\right)$ (viii) $\cot \left(\cos ^{-1} 3 / 5\right)$
$(vii)$ $(viii)$
Evaluate each of the following: (v) $\operatorname{Cosec}\left(\cos ^{-1} 8 / 17\right)$ (vi) $\operatorname{Sec}\left(\sin ^{-1} 12 / 13\right)$
(v) \[\begin{array}{*{35}{l}} {} \\ Let\text{ }co{{s}^{-1}}\left( 8/17 \right)\text{ }=\text{ }y \\ cos\text{ }y\text{ }=\text{ }8/17\text{ }where\text{ }y\text{ }\in \text{ }\left[ 0,\text{ }\pi...
Evaluate each of the following: (iii) $\operatorname{Sin}\left(\tan ^{-1} 24 / 7\right)$ (iv) $\operatorname{Sin}\left(\sec ^{-1} 17 / 8\right)$
(iii) (iv)
Evaluate each of the following: (i) $\sin \left(\sin ^{-1} 7 / 25\right)$ (ii) $\operatorname{Sin}\left(\cos ^{-1} 5 / 13\right)$
(i) \[\begin{array}{*{35}{l}} Given\text{ }sin\text{ }\left( si{{n}^{-1}}~7/25 \right) \\ let\text{ }y\text{ }=\text{ }si{{n}^{-1}}~7/25 \\ sin\text{ }y\text{ }=\text{ }7/25\text{ }where\text{...
Find the derivative of the function f defined by f (x) = mx + c at x = 0.
f(x) = mx + c, Checking the differentiability at x = 0 This is the derivative of a function at x = 0, and also this is the derivative of this function at every value of x.
If f (x) =${{x}^{3}}+7{{x}^{2}}+8x-9$, find f’ (4).
f(x) = x3 + 7x2 + 8x – 9, => Checking the differentiability at x = 4
If for the function Ø (x) =$\lambda {{x}^{2}}+7x-4$, Ø’ (5) = 97, find λ.
Finding the value of λ given in the real function and we are given with the differentiability of the function f(x) = λx2 + 7x – 4 at x = 5 which is f ‘(5) = 97 =>
Show that the derivative of the function f is given by f (x) = \[~\mathbf{2}{{\mathbf{x}}^{\mathbf{3}}}~\text{ }\mathbf{9}{{\mathbf{x}}^{\mathbf{2}~}}+\text{ }\mathbf{12}\text{ }\mathbf{x}\text{ }+\text{ }\mathbf{9}\], at x = 1 and x = 2 are equal.
We are given with a polynomial function f(x) = 2x3 – 9x2 + 12x + 9, and we have
If f is defined by f (x) = ${{x}^{2}}$ – 4x + 7, show that f’ (5) = 2 f’ (7/2)
Discuss the continuity and differentiability of the function f (x) = |x| + |x -1| in the interval of (-1, 2).
Since, a polynomial and a constant function is continuous and differentiable everywhere => f(x) is continuous and differentiable for x ∈ (-1, 0) and x ∈ (0, 1) and (1, 2). Checking continuity...
Show that the function $f$ is defined as follows $f(x)=\left\{\begin{array}{c} 3 x-2,02 \end{array}\right.$ Is continuous at $x=2$, but not differentiable thereat.
Since, LHL = RHL = f (2) Hence, F(x) is continuous at x = 2 Checking the differentiability at x = 2 $=> 5$ Since, (RHD at x = 2) ≠ (LHD at x = 2) Hence, f (2) is not differentiable at x =...
$\text { Show that } f(x)=\left\{\begin{array}{c} 12 x-13, \text { if } x \leq 3 \\ 2 x^{2}+5, \text { if } x>3 \end{array} \text { is dif ferentiable at } x=3 . \text { Also, find } f^{\prime}(3)\right.$
checking differentiability of given function at x = 3 => LHD (at x = 3) = RHD (at x = 3) = 12 Since, (LHD at x = 3) = (RHD at x = 3) Hence, f(x) is differentiable at x = 3.
Show that f (x) =${{x}^{\frac{1}{3}}}$ is not differentiable at x = 0.
Since, LHD and RHD does not exist at x = 0 Hence, f(x) is not differentiable at x = 0
Show that f (x) = |x – 3| is continuous but not differentiable at x = 3.
$\text { Dif ferentiate } \tan ^{-1}\left(\frac{1+a x}{1-a x}\right) \text { with respect to } \sqrt{1+a^{2} x^{2}} \text {. }$
$\text { Dif ferentiate } \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right) \text { with respect to } \cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right), \text { if } 0
$\text { Differentiate }(\cos x)^{\sin x} \text { with respect to }(\sin x)^{\cos x} .$
Differentiate $\sin ^{-1}\left(2 x \sqrt{1-x^{2}}\right)$ with respect to $\sec ^{-1}\left(\frac{1}{\sqrt{1-x^{2}}}\right)$, if, (i) $x \in(0,1 / \sqrt{2})$ (ii) $x \in(1 / \sqrt{2}, 1)$
(i) Let (ii) Let
$\text { Differentiate } \tan ^{-1}\left(\frac{\sqrt{1+x^{2}}-1}{x}\right) \text { with respect to } \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right), \text { if }-1
Differentiate $\sin ^{-1}\left(4 x \sqrt{1-4 x^{2}}\right)$ with respect to $\sqrt{1-4 x^{2}}$ if, (iii) $x \epsilon\left(-\frac{1}{2},-\frac{1}{2 \sqrt{2}}\right)$
(iii) Let
Differentiate $\sin ^{-1}\left(4 x \sqrt{1-4 x^{2}}\right)$ with respect to $\sqrt{1-4 x^{2}}$ if, (i) $x \epsilon\left(-\frac{1}{2 \sqrt{2}}, \frac{1}{2 \sqrt{2}}\right)$ (ii) $x \in\left(\frac{1}{2 \sqrt{2}}, \frac{1}{2}\right)$
(i) Let (ii)
Differentiate $\sin ^{-1} \sqrt{\left(1-x^{2}\right)}$ with respect to $\cos ^{-1} x$, if (i) $x \in(0,1)$ (ii) $x \in(-1,0)$
(i) Given sin-1 √ (1-x2) (ii) Given sin-1 √ (1-x2)
$\text { Differentiate }(\log x)^{\mathrm{x}} \text { with respect to } \log \mathrm{x} \text {. }$
$\text { Differentiate } \log \left(1+x^{2}\right) \text { with respect to } \tan ^{-1} x$
$\text { Differentiate } \mathbf{x}^{2} \text { with respect to } \mathrm{x}^{3}$
If $x=2 \cos \theta-\cos 2 \theta$ and $y=2 \sin \theta-\sin 2 \theta$, prove that $\frac{d y}{d x}=\tan \left(\frac{3 \theta}{2}\right)$.
Find dy/dx, when $x=\frac{1-t^{2}}{1+t^{2}} \text { and } y=\frac{2 t}{1+t^{2}}$
Find dy/dx, when $x=\cos ^{-1} \frac{1}{\sqrt{1+t^{2}}} \text { and } y=\sin ^{-1} \frac{1}{\sqrt{1+t^{2}}}, t \epsilon R$
Find dy/dx, when x = 2 t / 1+t^2 and y = 1-t^2 / 1+t^2.
Given, $x=2 t /\left(1+t^{2}\right)$ On differentiating $x$ with respect to t using quotient rule, $$ \begin{array}{l} \frac{\mathrm{dx}}{\mathrm{dt}}=\left[\frac{\left(1+\mathrm{t}^{2}\right)...
Find dy/dx, when $x=e^{\theta}\left(\theta+\frac{1}{\theta}\right) \text { and } y=e^{-\theta}\left(\theta-\frac{1}{\theta}\right)$
Find dy/dx, when $x=a(\cos \theta+\theta \sin \theta) \text { and } y=a(\sin \theta-\theta \cos \theta)$
Find dy/dx, when x = 3 a t / 1+t^2 and y = 3 a t^2/1+t^2
Find dy/dx, when $x=\frac{e^{t}+e^{-t}}{2} \operatorname{and} y=\frac{e^{t}-e^{-t}}{2}$
Find dy/dx, when $x=a(1-\cos \theta) \text { and } y=a(\theta+\sin \theta) \text { at } \theta=\pi / 2$
Find dy/dx, when $x=b \sin ^{2} \theta \text { and } y=a \cos ^{2} \theta$
Find dy/dx, when $x=a e^{\theta}(\sin \theta-\cos \theta), y=a e^{\theta}(\sin \theta+\cos \theta)$
Find dy/dx, when $x=a \cos \theta \text { and } y=b \sin \theta$
Find dy/dx, when$x=a(\theta+\sin \theta) \text { and } y=a(1-\cos \theta)$
Find dy/dx, when $x=a t^{2} \text { and } y=2 \text { at }$
$\text { If } y=\sqrt{\tan x+\sqrt{\tan x+\sqrt{\tan x+\ldots . \text { to } \infty}}}, \text { prove that } \frac{d y}{d x}=\frac{\sec ^{2} x}{2 y-1}$
$\text { If } y=\sqrt{\log x+\sqrt{\log x+\sqrt{\log x+\ldots . \text { to } \infty}}}, \text { prove that }(2 y-1) \frac{d y}{d x}=\frac{1}{x} \text {. }$
$\text { If } y=\sqrt{\cos x+\sqrt{\cos x+\sqrt{\cos x+\ldots . \text { to } \infty}}}, \text { prove that } \frac{d y}{d x}=\frac{\sin x}{1-2 y}$
$\text { If } y=\sqrt{x+\sqrt{x+\sqrt{x+\ldots . \text { to } \infty}}}, \text { prove that } \frac{d y}{d x}=\frac{1}{2 y-1}$
Differentiate the following functions with respect to x:$y=x^{n}+n^{x}+x^{x}+n^{n}$
Differentiate the following functions with respect to x:$y=e^{x}+10^{x}+x^{x}$
(vii) (viii)
Differentiate the following functions with respect to x:$(v)\left(x+\frac{1}{x}\right)^{x}+x^{\left(1+\frac{1}{x}\right)}$$\text { (vi) } e^{\sin x}+(\tan x)^{x}$
(v) (vi)
Differentiate the following functions with respect to x:$\text { (iii) } x^{x \cos x}+\frac{x^{2}+1}{x^{2}-1}$$(i v)(x \cos x)^{x}+(x \sin x)^{\frac{1}{x}}$
(iv)
Differentiate the following functions with respect to x:$x^{\tan ^{-1} x}$
Differentiate the following functions with respect to x:$(\tan x)^{1 / x}$
Differentiate the following functions with respect to x:$x^{\sin ^{-1} x}$
Differentiate the following functions with respect to x:$\left(\operatorname{Sin}^{-1} x\right)^{x}$
Differentiate the following functions with respect to x:$10^{\left(10^{x}\right)}$
Differentiate the following functions with respect to x:$(\log x)^{\log x}$
Differentiate the following functions with respect to x:$10^{\log \sin x}$
Differentiate the following functions with respect to x:$(\operatorname{Sin} x)^{\log x}$
Differentiate the following functions with respect to x:$e^{x \log x}$
Differentiate the following functions with respect to x:$(\operatorname{Sin} x)^{\cos x}$
Differentiate the following functions with respect to x:$(\log x)^{\cos x}$
Let y = (log x)cos x Taking log both the sides, we get
Differentiate the following functions with respect to x:$(\log x)^{x}$
Differentiate the following functions with respect to x:$x^{\cos ^{-1} x}$
Differentiate the following functions with respect to x:$(1+\cos x)^{x}$
Differentiate the following functions with respect to x:$x^{\sin x}$
Differentiate the following functions with respect to x:$x^{1 / x}$
If $x y^{2}=1$, prove that $2 \frac{d y}{d x}+y^{3}=0 .$
$\operatorname{If} x y=1$, prove that $\frac{d y}{d x}+y^{2}=0$.
If $y=\sqrt{1-x^{2}}+x \sqrt{1-y^{2}}=1$, prove that $\frac{d y}{d x}=\sqrt{\frac{1-y^{2}}{1-x^{2}}} .$
If $\sqrt{1-x^{2}}+\sqrt{1-y^{2}}=a(x-y)$, prove that $\frac{d y}{d x}=\sqrt{\frac{1-y^{2}}{1-x^{2}}}$
Find dy/dx in each of the following:$\operatorname{Sin} x y+\cos (x+y)=1$
differentiating the equation on both sides with respect to x, we get,
Find dy/dx in each of the following:$e^{x-y}=\log \left(\frac{x}{y}\right)$
Find dy/dx in each of the following: $\operatorname{tan}^{-1}\left(x^{2}+y^{2}\right)$
differentiating the equation on both sides with respect to x, we get,
Find dy/dx in each of the following: $\left(x^{2}+y^{2}\right)^{2}=x y$
differentiating the equation on both sides with respect to x, we get,
Find dy/dx in each of the following: $(x+y)^{2}=2 a x y$
differentiating the equation on both sides with respect to x, we get,
Find dy/dx in each of the following: $x^{5}+y^{5}=5 x y$
differentiating the equation on both sides with respect to x, we get,
Find dy/dx in each of the following: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$
Find dy/dx in each of the following: $4 x+3 y=\log (4 x-3 y)$
differentiating the equation on both sides with respect to x, we get,
Find dy/dx in each of the following:$\mathrm{x}^{2 / 3}+\mathrm{y}^{2 / 3}=\mathrm{a}^{2 / 3}$
differentiating the equation on both sides with respect to x, we get,
Find dy/dx in each of the following: $y^{3}-3 x y^{2}=x^{3}+3 x^{2} y$
differentiating the equation on both sides with respect to x, we get,
Find dy/dx in each of the following: $x y=c^{2}$
Differentiate the following functions with respect to x: $\cos ^{-1}\left\{\frac{1-x^{2 n}}{1+x^{2 n}}\right\}, 0
Differentiate the following functions with respect to x: $\sin ^{-1}\left\{\frac{1}{\sqrt{1+x^{2}}}\right\}$
Differentiate the following functions with respect to x: $\tan ^{-1}\left\{\frac{\sqrt{1+a^{2} x^{2}}-1}{a x}\right\}, x \neq 0$
Differentiate the following functions with respect to x: $\sin ^{-1}\left\{\frac{\sqrt{1+x}+\sqrt{1-x}}{2}\right\}, 0
Differentiate the following functions with respect to x: $\tan ^{-1}\left\{\frac{2 a^{x}}{1-a^{2 x}}\right\}, a>1,-\infty
Differentiate the following functions with respect to x: $\tan ^{-1}\left\{\frac{2^{x+1}}{1-4^{x}}\right\},-\infty
Differentiate the following functions with respect to x: $\tan ^{-1}\left\{\frac{4 x}{1-4 x^{2}}\right\},-\frac{1}{2}
Differentiate the following functions with respect to x: $\cos ^{-1}\left\{\frac{x+\sqrt{1-x^{2}}}{\sqrt{2}}\right\},-1
Differentiate the following functions with respect to x: $\sin ^{-1}\left\{\frac{x+\sqrt{1-x^{2}}}{\sqrt{2}}\right\},-1
Differentiate the following functions with respect to x: $\tan ^{-1}\left\{\frac{x}{a+\sqrt{a^{2}-x^{2}}}\right\},-a
Differentiate the following functions with respect to x: $\tan ^{-1}\left\{\frac{x}{1+\sqrt{1-x^{2}}}\right\},-1
Differentiate the following functions with respect to x:$\cos ^{-1}\left\{\frac{\cos x+\sin x}{\sqrt{2}}\right\},-\frac{\pi}{4}
Differentiate the following functions with respect to x: $\cos ^{-1}\left\{\frac{x}{\sqrt{x^{2}+a^{2}}}\right\}$
Differentiate the following functions with respect to x: ${Sin}^{-1}\left(1-2 x^{2}\right), 0
Differentiate the following functions with respect to x: ${Sin}^{-1}\left(2 x^{2}-1\right), 0
Differentiate the following functions with respect to x: $\sin ^{-1}\left\{\frac{x}{\sqrt{x^{2}+a^{2}}}\right\}$
Differentiate the following functions with respect to x: $ \tan ^{-1}\left\{\frac{x}{\sqrt{a^{2}-x^{2}}}\right\},-a
Differentiate the following functions with respect to x: $\sin ^{-1}\left\{\sqrt{1-x^{2}}\right\}, 0
Let,
Differentiate the following functions with respect to x: $\sin ^{-1}\left\{\sqrt{\frac{1-x}{2}}\right\}, 0
Let,
Differentiate the following functions with respect to x: $\text {} \cos ^{-1}\left\{\sqrt{\frac{1+x}{2}}\right\},-1
Differentiate the following functions with respect to x: $\text { 1. } \cos ^{-1}\left\{2 x \sqrt{1-x^{2}}\right\}, \frac{1}{\sqrt{2}}
Differentiate the following functions with respect to x: $\sqrt{{{\tan }^{-1}}(\frac{x}{2})}$
Differentiate the following functions with respect to x: ${{e}^{{{\tan }^{-1}}\sqrt{x}}}$
Differentiate the following functions with respect to x: ${{e}^{{{\sin }^{-1}}2x}}$
Differentiate the following functions with respect to x: ${{\tan }^{-1}}({{e}^{x}})$
Differentiate the following functions with respect to x: $\log (\frac{{{x}^{2}}+x+1}{{{x}^{2}}-x+1})$
Differentiate the following functions with respect to x: $\frac{{{e}^{2x}}+{{e}^{-2x}}}{{{e}^{2x}}-{{e}^{-2x}}}$
Differentiate the following functions with respect to x: log (cosec x – cot x)
Differentiate the following functions with respect to x: $\frac{{{e}^{x}}\log x}{{{x}^{2}}}$
Differentiate the following functions with respect to x: $\log (x+\sqrt{{{x}^{2}}+1})$
Differentiate the following functions with respect to x: x$\tan ({{e}^{\sin x}})$
Differentiate the following functions with respect to x: $\log \sqrt{\frac{1-\cos x}{1+\cos x}}$
Differentiate the following functions with respect to x: $\log (\frac{\sin s}{1+\cos x})$
Differentiate the following functions with respect to x: ${{e}^{\sqrt{\cot x}}}$
Differentiate the following functions with respect to x: ${{e}^{\tan 3x}}$
Differentiate the following functions with respect to x: ${{e}^{3x}}\cos 2x$
Differentiate the following functions with respect to x: $\sin (\frac{1+{{x}^{2}}}{1-{{x}^{2}}})$
Differentiate the following functions with respect to x: $\sqrt{\frac{1+x}{1-x}}$
Differentiate the following functions with respect to x: ${{(\log \sin x)}^{2}}$.
Let y = (log sin x)2
Differentiate the following functions with respect to x: $\sqrt{\frac{1+{{x}^{2}}}{1-{{x}^{2}}}}$
Differentiate the following functions with respect to x: $\sqrt{\frac{1+\sin x}{1-\sin x}}$
Differentiate the following functions with respect to x: ${{3}^{x\log x}}$
Differentiate the following functions with respect to x: ${{3}^{{{x}^{2}}+2x}}$
Differentiate the following functions with respect to x: \[\mathbf{lo}{{\mathbf{g}}_{\mathbf{x}}}~\mathbf{3}\]
Differentiate the following functions with respect to x: ${{3}^{{{e}^{x}}}}$
Differentiate the following functions with respect to x: ${{2}^{{{x}^{3}}}}$
Differentiate the following functions with respect to x: tan 5x
Let y = tan (5x°)
Differentiate the following functions with respect to x: \[\mathbf{lo}{{\mathbf{g}}_{\mathbf{7}}}~\left( \mathbf{2x}\text{ }\text{ }\mathbf{3} \right)\]
Differentiate the following functions with respect to x: \[~\mathbf{Si}{{\mathbf{n}}^{\mathbf{2}}}~\left( \mathbf{2x}\text{ }+\text{ }\mathbf{1} \right)\]
Let y = sin2 (2x + 1) On differentiating y with respect to x, we get
Differentiate the following functions with respect to x: ${{e}^{\tan{x}}}$
Differentiate the following functions with respect to x: ${{e}^{\sin \sqrt{x}}}$
Differentiate the following functions with respect to x: Sin (log x)
Differentiate the following functions with respect to x: \[\mathbf{tan}\text{ }({{\mathbf{x}}^{\mathbf{o}}}~+\text{ }\mathbf{4}{{\mathbf{5}}^{\mathbf{o}}})\]
Differentiate the following functions with respect to x: ${{\tan }^{2}}x$
Given tan2 x
Differentiate the following functions from the first principles: $e^{\sqrt{2 x}}$
Differentiate the following functions from the first principles: $e^{cosx}$
let f (x) = ecos x By using the first principle formula, we get,
Differentiate the following functions from the first principles: $e^{ax+b}$
Differentiate the following functions from the first principles: $e^{-x}$
$\text { Discuss the continuity of the function } f(x)=\left\{\begin{array}{c} 2 x-1, \text { if } x<2 \\ \frac{3 x}{2}, \text { if } x \geq 2 \end{array}\right.$
A real function f is said to be continuous at x = c, where c is any point in the domain of f if A function is continuous at x = c if Function is changing its nature (or expression) at x = 2, so we...
If $f(x)=\frac{\tan \left(\frac{\pi}{4}-x\right)}{\cot 2 x}$ for $x \neq \pi / 4$, find the value which can be assigned to $f(x)$ at $x=\pi / 4$ so that the function $f(x)$ becomes continuous everywhere in $[0, \pi / 2]$.
A real function f is said to be continuous at x = c, where c is any point in the domain of f if Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as...
The function $f(x)$ is defined by $f(x)=\left\{\begin{array}{c}x^{2}+a x+b, 0 \leq x<2 \\ 3 x+2,2 \leq x \leq 4 \\ 2 a x+5 b, 4
A real function f is said to be continuous at x = c, where c is any point in the domain of f if h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c...
Find the values of a and b so that the function f (x) defined by $f(x)=\left\{\begin{array}{ll}x+a \sqrt{2} \sin x, & \text { if } 0 \leq x<\frac{\pi}{4} \\ 2 x \cot x+b, & \text { if } \frac{\pi}{4} \leq x<\frac{\pi}{2} \\ a \cos 2 x-b \sin x, & \text { if } \frac{\pi}{2} \leq x<\pi\end{array}\right.$
The function $f(x)=\left\{\begin{array}{c}\frac{x^{2}}{a}, \text { if } 0 \leq x<1 \\ a, \text { if } 1 \leq x<\sqrt{2} \\ \frac{2 b^{2}-4 b}{x^{2}}, \text { if } \sqrt{2} \leq x<\infty\end{array}\right.$ Is continuous on $[0, \infty)$. Find the most suitable values of $a$ and $b$.
A real function f is said to be continuous at x = c, where c is any point in the domain of f if h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c...
In the following, determine the value(s) of constant(s) involved in the definition so that the given function is continuous: (iii) $f(x)=\left\{\begin{array}{c}k\left(x^{2}+3 x\right) \text { if } x<0 \\ \cos 2 x, \text { if } x \geq 0\end{array}\right.$ (i v) $f(x)=\left\{\begin{array}{c}2 \text { if } x \leq 3 \\ a x+b, \text { if } 3
(iii) A real function f is said to be continuous at x = c, where c is any point in the domain of f if h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x →...
In the following, determine the value(s) of constant(s) involved in the definition so that the given function is continuous: (i) $f(x)=\left\{\begin{array}{cc}\frac{\sin 2 x}{5 x} \text { if } x \neq 0 \\ 3 k, \quad i f x=0\end{array}\right.$ (ii) $f(x)=\left\{\begin{array}{l}k x+5 \text { if } x \leq 2 \\ x-1, \text { if } x>2\end{array}\right.$
(i) A real function f is said to be continuous at x = c, where c is any point in the domain of f if h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c...
Find the points of discontinuity, if any, of the following functions: (x i) $f(x)=\left\{\begin{array}{c}2 x, \text { if } x<0 \\ 0, \text { if } 0 \leq x \leq 1 \\ 4 x, \text { if } x>1\end{array}\right.$ (x i i) $f(x)=\left\{\begin{array}{c}\sin x-\cos x, \text { if } x \neq 0 \\ -1, \text { if } x=0\end{array}\right.$
(xi) A real function f is said to be continuous at x = c, where c is any point in the domain of f if h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x →...
Find the points of discontinuity, if any, of the following functions: (i x) $f(x)=\left\{\begin{array}{c}|x|+3, \text { if } x \leq-3 \\ -2 x, \text { if }-33\end{array}\right.$ (x) $f(x)=\left\{\begin{array}{c}x^{10}-1, \text { if } x \leq 1 \\ x^{2}, \text { if } x>1\end{array}\right.$
(ix) A real function f is said to be continuous at x = c, where c is any point in the domain of f if h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x →...
Find the points of discontinuity, if any, of the following functions: $(v i i) f(x)=\left\{\begin{array}{c}\frac{e^{x}-1}{\log _{c}(1+2 x)}, \text { if } x \neq 0 \\ 7, \text { if } x=0\end{array}\right.$ (viii) $f(x)=\left\{\begin{array}{c}|x-3|, \text { if } x \geq 1 \\ \frac{x^{2}}{4}-\frac{3 x}{2}+\frac{13}{4}, \text { if } x<1\end{array}\right.$
((vii) A real function f is said to be continuous at x = c, where c is any point in the domain of f if h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x...
Find the points of discontinuity, if any, of the following functions: (i) $f(x)=\left\{\begin{array}{c}x^{3}-x^{2}+2 x-2, \text { if } x \neq 1 \\ 4, \text { if } x=1\end{array}\right.$ (ii) $f(x)=\left\{\begin{array}{l}\frac{x^{4}-16}{x-2}, \text { if } x \neq 2 \\ 16, \text { if } x=2\end{array}\right.$
(i) A real function f is said to be continuous at x = c, where c is any point in the domain of f if h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c...
Discuss the the continuity of the function $f(x)=\left\{\begin{array}{c}\frac{x}{|x|}, x \neq 0 \\ 0, x=0\end{array}\right.$
A real function f is said to be continuous at x = c, where c is any point in the domain of f if Since, h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x...
Prove that the function $f(x)=\left\{\begin{array}{c}\frac{\sin x}{x}, x<0 \\ x+1, x \geq 0\end{array}\right.$ is everywhere continuous.
A real function f is said to be continuous at x = c, where c is any point in the domain of f if A function is continuous at x = c if From definition of f(x), f(x) is defined for all real numbers....
Find the inverse of the following matrices by using elementary row transformations: $\left[\begin{array}{cc}7 & 1 \\ 4 & -3\end{array}\right]$
Solution: For row transformation $\begin{array}{l} A=I A \\ \Rightarrow\left[\begin{array}{cc} 7 & 1 \\ 4 & -3 \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1...
43.$\frac{(\cos ecA)}{(\cos ecA-1)}+\frac{(\cos ecA)}{(\cos ecA+1)}=2{{\sec }^{2}}A$
Assuming the L.H.S and taking L.C.M and on simplifying we will get, $=\frac{(\cos ecA)(\cos ecA+1+\cos ecA-1)}{(\cos e{{c}^{2}}A-1)}$ $=\frac{(2\cos e{{c}^{2}}A)}{{{\cot }^{2}}A}$ $=\frac{2{{\sin...
42.$\frac{\cos A}{1-\tan A}+\frac{\sin A}{1-\cot A}=\sin A+\cos A$
Solving the L.H.S, we will get $=\frac{\cos A}{1-\tan A}+\frac{\sin A}{1-\cot A}$ $=\frac{\cos A}{1-\frac{\sin A}{\cos A}}+\frac{\sin A}{1-\frac{\cos A}{\sin A}}$ $=\frac{{{\cos }^{2}}A}{\cos A-\sin...
41.$\frac{1-\cos A}{1+\cos A}={{(\cot A-\cos ecA)}^{2}}$
Assuming L.H.S and taking L.C.M and on simplifying we will get, $=\frac{\sec A+1+\sec A-1}{(\sec A+1)(\sec A-1)}$ $=\frac{2\sec A}{({{\sec }^{2}}A-1)}$ $=\frac{2{{\cos }^{2}}A}{(\cos A{{\sin...
40. $\frac{1-\cos A}{1+\cos A}={{(\cot A-\cos ecA)}^{2}}$
Solving L.H.S and divide the numerator and denominator with $(1-\cos A),$, we have $=\frac{(1-\cos A)(1-\cos A)}{(1+\cos A)(1-\cos A)}$ $=\frac{{{(1-\cos A)}^{2}}}{(1+{{\cos }^{2}}A)}$...
39.${{(\sec A-\tan A)}^{2}}=\frac{1-\sin A}{1+\sin A}$
Solving $LHS={{(\sec A-\tan A)}^{2}}$, we get $={{\left[ \frac{1}{\cos A}-\frac{\sin A}{\cos A} \right]}^{2}}$ $=\frac{{{(1-\sin A)}^{2}}}{{{\cos }^{2}}A}$ $=\frac{{{(1-\sin A)}^{2}}}{1-{{\sin...
38. Prove that:(iii)$$$\sqrt{\frac{(1+\cos \theta )}{(1-\cos \theta )}}+\sqrt{\frac{(1-\cos \theta )}{(1+\cos \theta )}}=2\cos ec\theta $(iv) $\frac{\sec \theta -1}{\sec \theta -1}={{\left( \frac{\sin \theta }{1+\cos \theta } \right)}^{2}}$
Solving L.H.S and dividing the numerator and denominator with its respective conjugates, we have $=\sqrt{\frac{(1-\cos \theta )(1-\cos \theta )}{(1+\cos \theta )(1-\cos \theta...
38. Prove that: (i)$\sqrt{\frac{(\sec \theta -1)}{(\sec \theta +1}}+\sqrt{\frac{(\sec \theta +1)}{(\sec \theta -1)}}=2\cos ec\theta $(ii)$\sqrt{\frac{(1+\sin \theta )}{(1-\sin \theta )}}+\sqrt{\frac{(1\sin \theta )}{(1+\sin \theta )}}=2\sec \theta $
Solving L.H.S and divide the numerator and denominator with its respective conjugates, we have $=\sqrt{\frac{(\sec \theta -1)(\sec \theta -1)}{(\sec \theta +1)(\sec \theta -1)}}+\sqrt{\frac{(\sec...
37. (i) $\sqrt{\frac{1+\sin A}{1-\sin A}}=\sec A+\tan A$(ii)$\sqrt{\frac{(1-\cos A)}{(1+\cos A)}}+\sqrt{\frac{(1+\cos A)}{(1-\cos A)}}=2\cos ecA$
Solving L.H.S and dividing the numerator and denominator with $\sqrt{(1+\sin A)},$we have $=\sqrt{\frac{(1+\sin A)(1+\sin A)}{(1-\sin A)(1+\sin A)}}=\sqrt{\frac{{{(1+\sin A)}^{2}}}{1-{{\sin...
36.$1+\frac{\cos A}{\sin A}=\frac{\sin A}{1-\cos A}$
Solving L.H.S $LHS=\frac{1+\cos A}{\sin A}$ Multiply the numerator and denominator by $(1-\cos A)$ we will have $=\frac{(1+\cos A)(1-\cos A)}{\sin A(1-\cos A)}$ $=\frac{1-{{\cos }^{2}}A}{\sin...
35.$\frac{\sec A-\tan A}{\sec A+\tan A}=\frac{{{\cos }^{2}}A}{{{(1+\sin A)}^{2}}}$
Solving L.H.S $LHS=\frac{\sec A-\tan A}{\sec A+\tan A}$ Dividing the denominator and numerator with $(\sec A+\tan A)$ and using ${{\sec }^{2}}\theta -{{\tan }^{2}}\theta =1,$we have, $=\frac{{{\sec...
34. $\frac{1+\cos A}{{{\sin }^{2}}A}=\frac{1}{1-\cos A}$
Solving L.H.S and using the trigonometric identity ${{\sin }^{2}}A+{{\cos }^{2}}A=1,$, we have ${{\sin }^{2}}A=1-{{\cos }^{2}}A$ $\Rightarrow {{\sin }^{2}}A=(1-\cos A)(1+\cos A)$ $LHS=\frac{1+\cos...
Find the adjoint of each of the following matrices:
(i) $\left[\begin{array}{cc}-3 & 5 \\ 2 & 4\end{array}\right]$
(ii) $\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$ Verify that $(\operatorname{adj} A) A=|A| I=A(\operatorname{adj} A)$ for the above matrices.
Solution: (i) Suppose $A=\left[\begin{array}{cc}-3 & 5 \\ 2 & 4\end{array}\right]$ Cofactors of $A$ are $C_{11}=4$ $C_{12}=-2$ $C_{21}=-5$ $C_{22}=-3$ Since, adj...
33.$\frac{(1+{{\tan }^{2}}\theta )\cot \theta }{\cos e{{c}^{2}}\theta }=\tan \theta $
First solve L.H.S and using the trigonometric identity we all know that ${{\sec }^{2}}\theta {{\tan }^{2}}\theta =1\Rightarrow 1+{{\tan }^{2}}\theta ={{\sec }^{2}}\theta $ $LHS=\frac{{{\sec...
32. $\cos e{{c}^{6}}\theta ={{\cot }^{6}}\theta +3{{\cot }^{2}}\theta \cos e{{c}^{2}}\theta +1$
Using the trigonometric identity we get $\cos e{{c}^{2}}\theta +{{\cot }^{2}}\theta =1$ cubing it on both side ${{(\cos e{{c}^{6}}\theta +{{\cot }^{2}}\theta )}^{3}}=1$ $\cos e{{c}^{6}}-{{\cot...
31. ${{\sec }^{2}}\theta ={{\tan }^{2}}\theta +3{{\tan }^{2}}\theta {{\sec }^{2}}\theta +1$
Using trigonometric identity, ${{\sec }^{2}}\theta -{{\tan }^{2}}\theta =1$ Cubing it on both side ${{({{\sec }^{2}}\theta -{{\tan }^{2}}\theta )}^{2}}=1$ ${{\sec }^{6}}\theta -{{\tan }^{6}}\theta...
30. $\frac{\tan \theta }{1-\cot \theta }+\frac{\cot \theta }{1-\tan \theta }=1+\tan \theta +\cot \theta $
Solving L.H.S, we get $LHS=\frac{\tan \theta }{1-\frac{1}{\tan \theta }}+\frac{\cot \theta }{1-\tan \theta }$ $=\frac{{{\tan }^{2}}\theta }{\tan \theta -1}+\frac{\cot \theta }{1-\tan \theta }$...
29. $\frac{1+\sec \theta }{\sec \theta }=\frac{{{\sin }^{2}}\theta }{1-\cos \theta }$
Solving L.H.S and using the trigonometric identity ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1,$we get Multiplying by $(1-\cos \theta )$to Numerator and denominator $LHS=\frac{1+\sec \theta }{\sec...
28.$\frac{1+{{\tan }^{2}}\theta }{1+{{\cot }^{2}}\theta }={{\tan }^{2}}\theta $
Solve L.H.S $\frac{1+{{\tan }^{2}}\theta }{1+{{\cot }^{2}}\theta }$ Using trigonometric identity${{\sec }^{2}}\theta -{{\tan }^{2}}\theta =1,and\cos e{{c}^{2}}\theta -{{\cot }^{2}}\theta =1$...
27.$\frac{{{(1+\sin \theta )}^{2}}+{{(1-\sin \theta )}^{2}}}{2{{\cos }^{2}}\theta }=\frac{1+{{\sin }^{2}}\theta }{1-{{\sin }^{2}}\theta }$
Firstly we will solve L.H.S=R.H.S Then use trigonometric identity $\sin \theta +\cos \theta =1,$, we get $LHS=\frac{{{(1+\sin \theta )}^{2}}+{{(1-\sin \theta )}^{2}}}{2{{\sec }^{2}}\theta }$...
26. $\frac{1+\sin \theta }{\cos \theta }+\frac{\cos \theta }{1+\sin \theta }=2\sec \theta $
Firstly we will solve L.H.S Using the trigonometric identity ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1,$, we get $LHS=\frac{1+\sin \theta }{\cos \theta }+\frac{\cos \theta }{1+\sin \theta }$...
25. $\frac{1}{1+\sin A}+\frac{1}{1-\sin A}=2{{\sec }^{2}}A$
Firstly we will solve L.H.S $LHS=\frac{1}{1+\sin A}+\frac{1}{1-\sin A}$ $=\frac{(1-\sin A)+(1+\sin A)}{(1+\sin A)(1-\sin A)}$ $=\frac{1-\sin A+1+\sin A}{1-{{\sin }^{2}}A}$ $\because (1+\sin...
Find the principal value of the following:
(i) $\sin ^{-1}\left(\cos \frac{3 \pi}{4}\right)$
(ii) $\sin ^{-1}\left(\tan \frac{5 \pi}{4}\right)$
Solution: (i) Suppose $\sin ^{-1}\left(\cos \frac{3 \pi}{4}\right)=\mathrm{y}$ Therefore we can write the above equation as $\sin \mathrm{y}=\cos \frac{3 \pi}{4}=-\sin \left(\pi-\frac{3...
Find the principal value of the following:
(i) $\sin ^{-1}\left(\frac{\sqrt{3}-1}{2 \sqrt{2}}\right)$
(ii) $\sin ^{-1}\left(\frac{\sqrt{3}+1}{2 \sqrt{2}}\right)$
Solution: (i) It is given that functions can be written as $\sin ^{-1}\left(\frac{\sqrt{3}-1}{2 \sqrt{2}}\right)=\sin ^{-1}\left(\frac{\sqrt{3}}{2 \sqrt{2}}-\frac{1}{2 \sqrt{2}}\right) $Taking $1 /...