Solution: The provided expression is $y=e^{a c o s^{-} x}$ $\frac{d y}{d x}=e^{a \cos ^{-1} x} \cdot \frac{d}{d x} a \cos ^{-1} x$ $=e^{a \operatorname{acs}^{-1 x} x} \cdot...
If $y=\left|\begin{array}{ccc}f(x) & g(x) & h(x) \\ l & m & n \\ a & b & c\end{array}\right|$ prove that $\frac{d y}{d x}=\left|\begin{array}{ccc}f^{\prime}(x) & g^{\prime}(x) & h^{\prime}(x) \\ l & m & n \\ a & b & c\end{array}\right|$
Solution: The provided expression is $y=\left|\begin{array}{ccc}f(x) & g(x) & h(x) \\ l & m & n \\ a & b & c\end{array}\right|$ Now applying the derivative: $\frac{d y}{d...
Does there exist a function which is continuous everywhere but not differentiable at exactly two points?
Solution: Let's consider the function $f(x)=|x|+|x-1|$ The $f$ is continuous everywhere but it isn't differentiable at $x=0$ and $x=1$.
Using the fact that $\sin (A+B)=\sin A \cos B+\cos A \sin B$ and the differentiation, obtain the sum formula for cosines.
Solution: The provided expression is $\sin (A+B)=\sin A \cos B+\cos A \sin B$ Let's consider $A$ and $B$ as function of $t$ and differentiating both sides with respect to $x$, $\cos...
Using mathematical induction, prove that $\frac{d}{d x}\left(x^{n}\right)=n x^{n-1}$ for all positive integers $n$.
Solution: Let's consider the given statement be $p(n)$. $p(n)=\frac{d}{d x}\left(x^{n}\right)=n x^{n-1} \ldots \dots \dots$(1) Step 1: At $n=1$ result is true $p(1)=\frac{d}{d...
If $f(x)=|x|^{3}$, show that $f^{\prime \prime}(x)$ exists for all real $x$ and find it.
Solution: The provided expression is $f(x)=\left|x^{3}\right|_{=}\left\{\begin{array}{lll}x^{3},&\text {if } x \geq 0 \\ \left(-x^{3}\right), & \text {if }\quad x<0\end{array}\right.$...
If $x=a(\cos t+t \sin t)$ and $y=a(\sin t-t \cos t)$, find $\frac{d^{2} y}{d x^{2}}$.
Solution: The provided expressions are $x=a(\cos t+t \sin t)$and $y=a(\sin t-t \cos t)$ $x=a(\cos t+t \sin t)$ With respect to $t$ differentiating both the sides $\frac{d x}{d t}=a\left(-\sin...
If ${\cos y=x \cos (a+y)}$ with $\cos a \neq \pm 1$, prove that $\frac{d y}{d x}=\frac{\cos ^{2}(a+y)}{\sin a}$.
Solution: The provided expression is $\cos y=x \cos (a+y)$ $x=\frac{\cos y}{\cos (a+y)}$ With respect to $y$ apply derivative $\frac{d x}{d y}=\frac{d}{d y}\left(\frac{\cos y}{\cos (a+y)}\right)$...
If $(x-a)^{2}+(y-b)^{2}=c^{2}=$ for some $c>0$, prove that $\frac{\left[1+\left(\frac{d y}{d x}\right)^{2}\right]^{\frac{3}{2}}}{\frac{d^{2} y}{d x^{2}}}$ is a constant independent of $\mathbf{a}$ and $\mathbf{b}$.
Solution: The provided expression is $(x-a)^{2}+(y-b)=c^{2}$.........(1) Now applying derivation: $2(x-a)+2(y-b) \frac{d y}{d x}=0$ $2(x-a)=-2(y-b) \frac{d y}{d x}$ $\frac{d y}{d...
If $x \sqrt{1+y}+y \sqrt{1+x}=0$, for $-1<x<1$, Prove that $\frac{d y}{d x}=\frac{-1}{(1+x)^{2}}$.
Solution: The provided expression is $x \sqrt{1+y}+y \sqrt{1+x}=0$ $x \sqrt{1+y}=-y \sqrt{1+x}$ Square both the sides: $x^{2}(1+y)=y^{2}(1+x)$ $x^{2}+x^{2} y=y^{2}+y^{2} x$ $x^{2}-y^{2}=-x^{2}...
Find $\frac{d y}{d x}$ if $y=\sin ^{-1} x+\sin ^{-1} \sqrt{1-x^{2}},-1 \leq x \leq 1$.
Solution: The provided expression is $y=\sin ^{-1} x+\sin ^{-1} \sqrt{1-x^{2}}$ Now applying derivation: $\frac{d y}{d x}=\frac{1}{\sqrt{1-x^{2}}}+\frac{1}{\sqrt{1-\left(\sqrt{1-x^{2}}\right)^{2}}}...
Find $\frac{d y}{d x}$ if $y=12(1-\cos t)$ and $x=10(t-\sin t),-\frac{\pi}{2}<t<\frac{\pi}{2}$.
Solution: The provided expressions are $y=12(1-\cos t)$ and $x=10(t-\sin t)$ $\frac{d y}{d t}=12 \frac{d}{d t}(1-\cos t)=12(0+\sin t)=12 \sin t$ and $\frac{d x}{d t}=10 \frac{d}{d t}(1-\cos t)$...
Differentiate with respect to $x$ the functions in exercise $x^{x^{2}-3}+(x-3)^{x^{2}} \text { for } x>3$
Solution: Let's consider $y=x^{x^{2}-3}+(x-3)^{x^{2}}$ For the value $x>3$ Putting the value of $u=x^{x^{2}-3} \text { and } v=(x-3)^{x^{2}}$ $\frac{d y}{d x}=\frac{d u}{d x}+\frac{d v}{d x}...
Differentiate with respect to $x$ the functions in exercise $x^{x}+x^{2}+a^{x}+a^{a}$, for some fixed $a>0$ and $x>0$.
Solution: Let's consider $y=x^{x}+x^{a}+a^{x}+a^{a}$ Now applying derivation: $\frac{d y}{d x}=\frac{d}{d x} x^{x}+\frac{d}{d x} x^{a}+\frac{d}{d x} a^{x}+\frac{d}{d x} a^{a}$ $=\frac{d}{d x}...
Differentiate with respect to $x$ the functions in exercise $(\sin x-\cos x)^{\sin x-\cos x}$: $\frac{\pi}{4}<x<\frac{3 \pi}{4}$
Solution: Let's Consider $y=(\sin x-\cos x)^{\sin x-\cos x} \text {........ }$(i) $=(\sin x-\cos x) \log (\sin x-\cos x)$ Now applying derivation: $\frac{d}{d x} \log y=(\sin x-\cos x) \frac{d}{d...
Differentiate with respect to $x$ the functions in exercise $\cos (a \cos x+b \sin x)$ for some constants $a$ and $b$
Solution: Let's consider $y=\cos (a \cos x+b \sin x)$ for some constants $a$ and $b$ Now applying derivation: $\frac{d y}{d x}=-\sin (a \cos x+b \sin x) \frac{d}{d x}(a \cos x+b \sin x)$ $\frac{d...
Differentiate with respect to $x$ the functions in exercise $(\log x)^{\log x}, x>1$
Solution: Let's consider $y=(\log x)^{\operatorname{bgg} x}, x>1 \text {............ }$(i) On both the sides taking log: $\log y=\log (\log x)^{\log x}=\log x \log (\log x)$ Now applying...
Differentiate with respect to $x$ the functions in exercise $\cot ^{-1}\left[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right], 0<x<\frac{\pi}{2}$
Solution: Let's consider $y=\cot ^{-1}\left(\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right), 0<x<\frac{\pi}{2}$.............(i) Simplify the function,...
Differentiate with respect to $x$ the functions in Exercise $\sin ^{-1}(x \sqrt{x}), 0 \leq x \leq 1$
Solution: Now let's consider $y=\sin ^{-1}(x \sqrt{x})$ or $y=\frac{\sin ^{-1}\left(x^{\frac{3}{2}}\right)}{}$ Now derivating the above given function: $\frac{d y}{d...
Differentiate with respect to $x$ the functions in Exercise $(5 x)^{3 \cos 2 x}$
Solution: Now let's consider $y=(5 x)^{3 \cos 22}$ On both the sides taking log, we obtain $\log y=\log (5 x)^{3 \cos 2 x}$ $\log y=3 \cos 2 x \log (5 x)$ Now derivating the above given function:...
Differentiate with respect to $x$ the functions in Exercise $\sin ^{3} x+\cos ^{6} x$
Solution: Now let's consider $y=\sin ^{3} x+\cos ^{6} x$ or $y=(\sin x)^{3}+(\cos x)^{6}$ $\frac{d y}{d x}=3(\sin x)^{2} \frac{d}{d x} \sin x+6(\cos x)^{3} \frac{d}{d x} \cos x$ $\frac{d y}{d x}=3...
Differentiate with respect to $x$ the functions in Exercise $\left(3 x^{2}-9 x+5\right)^{9}$
Solution: Now let's consider $y=\left(3 x^{2}-9 x+5\right)^{2}$ $\frac{d y}{d x}=9\left(3 x^{2}-9 x+5\right)^{8} \frac{d}{d x}\left(3 x^{2}-9 x+5\right)$ $\left[\because \frac{d}{d...