Solution: Given that $f(x)=x^{9}+4 x^{7}+11$ Now differentiating above equation w.r.t $x$, we obtain $\begin{array}{l} \Rightarrow \\ f(x)=\frac{d}{d x}\left(x^{9}+4 x^{7}+11\right) \\ \Rightarrow...

Solution: Given that $f(x)=x^{2}-x+1$ By differentiating the given equation with respect to $x$, we obtain $\Rightarrow \mathrm{t}$ $\mathrm{f}(\mathrm{x})=\frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{x}$...

Solution: Given that $f(x)=(x-1) e^{x}+1$ Now differentiating the given eq. with respect to $x$, we obtain $\begin{array}{l} f(x)=\frac{d}{d x}\left((x-1) e^{x}+1\right) \\ \Rightarrow...

Solution: Given that, $f(x)=\sin \left(2 x+\frac{\pi}{4}\right)$ $\begin{array}{l} \Rightarrow f^{\prime}(x)=\frac{d}{d x}\left\{\sin \left(2 x+\frac{\pi}{4}\right)\right\} \\ \Rightarrow f(x)=\cos...

Solution: $\begin{array}{l} \text { Given that } f(x)=\tan ^{-1}(\sin x+\cos x) \\ \Rightarrow f^{\prime}(x)=\frac{d}{d x}\left(\tan ^{-1}(\sin x+\cos x)\right) \\ \Rightarrow f(x)=\frac{1}{1+(\sin...

Solution: Given that $f(x)=\tan x$ $\begin{array}{l} \Rightarrow \\ f(x)=\frac{d}{d x}(\tan x) \\ \Rightarrow f^{\prime}(x)=\sec ^{2} x \end{array}$ As given $x \in(-\pi / 2, \pi / 2)$ i.e.,...

Solution: Given that $f(x)=\cos x$ $\Rightarrow$ $\begin{array}{l} f(x)=\frac{d}{d x}(\cos x) \\ \Rightarrow f^{\prime}(x)=-\sin x \end{array}$ Taking different region from 0 to $2 \pi$ Suppose...

Solution: Given that $f(x)=\sin x$ $\begin{array}{l} \Rightarrow \\ f(x)=\frac{d}{d x}(\sin x) \\ \Rightarrow f^{\prime}(x)=\cos x \end{array}$ As given $x \in(-\pi / 2, \pi / 2)$. i.e. $4^{\text...

Solution: Given that $f(x)=\cos ^{2} x$ $\begin{array}{l} f(x)=\frac{d}{d x}\left(\cos ^{2} x\right) \\ \Rightarrow f^{\prime}(x)=2 \cos x(-\sin x) \\ \Rightarrow f^{\prime}(x)=-2 \sin (x) \cos (x)...

Solution: Given that$f(x)=x^{3}-15 x^{2}+75 x-50$ $\Rightarrow$ $f(x)=\frac{d}{d x}\left(x^{3}-15 x^{2}+75 x-50\right)$ $\Rightarrow f^{\prime}(x)=3 x^{2}-30 x+75$ $\Rightarrow...

Solution: Given that $f(x)=x-\sin x$ $\Rightarrow$ $\begin{array}{l} f(x)=\frac{d}{d x}(x-\sin x) \\ \Rightarrow f^{\prime}(x)=1-\cos x \end{array}$ As given $x \in R$ $\begin{array}{l}...

Solution: Given that $f(x)=\log \sin x$ $\begin{array}{l} \Rightarrow f^{\prime}(x)=\frac{d}{d x}(\log \sin x) \\ \Rightarrow f^{\prime}(x)=\frac{1}{\sin x} \times \cos x \\ \Rightarrow...

Solution: It is given that $f(x)=\sin x$ $\begin{array}{l} \Rightarrow f(x)=\frac{d}{d x}(\sin x) \\ \Rightarrow f^{\prime}(x)=\cos x \end{array}$ Taking different region from 0 to $2 \pi$ we get...

Solution: $\begin{array}{l} \text { Given } f(x)=\log _{a} x, 0<a<1 \\ \Rightarrow f(x)=\frac{d}{d x}(\log a x) \\ \Rightarrow f(x)=\frac{1}{x l o g a} \end{array}$ As it is given...

Solution: Given that $\mathrm{f}(\mathrm{x})=\mathrm{e}^{\frac{1}{\mathrm{x}}}$ $\begin{array}{l} \Rightarrow...

Solution: It is given that $f(x)=e^{2 x}$ $\Rightarrow$ $\begin{array}{l} \mathrm{f}(\mathrm{x})=\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{e}^{2 \mathrm{x}}\right) \\ \Rightarrow...

Solution: $\begin{array}{l} \text { Given } f(x)=\sin x-\cos x \\ \Rightarrow f^{\prime}(x)=\frac{d}{d x}(\sin x-\cos x) \\ \Rightarrow f^{\prime}(x)=\cos x+\sin x \end{array}$ For $f(x)$ let's find...

Solution: Given that $f(x)=x^{2}-6 x+9$ $\begin{array}{l} \Rightarrow \\ f(x)=\frac{d}{d x}\left(x^{2}-6 x+9\right) \\ \Rightarrow f^{\prime}(x)=2 x-6 \\ \Rightarrow f^{\prime}(x)=2(x-3)...

Solution: (i) Given that $f(x)=2 x^{3}-15 x^{2}+36 x+1$ Now with respect to $x$ differentiating above equation, we obtain $\begin{array}{l} f(x)=\frac{d}{d x}\left(2 x^{3}-15 x^{2}+36 x+1\right) \\...

Solution: (i) Given that $f(x)=5 x^{3}-15 x^{2}-120 x+3$ Now with respect to $x$ differentiating above equation, we obtain $\begin{array}{l} f^{\prime}(x)=\frac{d}{d x}\left(5 x^{3}-15 x^{2}-120...

Solution: (i) Given that $f(x)=5+36 x+3 x^{2}-2 x^{3}$ $f(x)=\frac{d}{d x}\left(5+36 x+3 x^{2}-2 x^{3}\right)$ $\Rightarrow f^{\prime}(x)=36+6 x-6 x^{2}$ For $f(x)$ now we have to find critical...

Solution: (i) Given that $f(x)=6-9 x-x^{2}$ $\begin{array}{l} \Rightarrow f^{\prime}(x)=\frac{d}{d x}\left(6-9 x-x^{2}\right) \\ \Rightarrow f^{\prime}(x)=-9-2 x \end{array}$ For $f(x)$ to be...

Solution: (i) Given that $f(x)=10-6 x-2 x^{2}$ On differentiating the above equation we obtain, $\begin{array}{l} \Rightarrow f^{\prime}(x)=\frac{d}{d x}\left(10-6 x-2 x^{2}\right) \\ \Rightarrow...

Solution: It is given that, $f(x)=a x+b, a<0$ Suppose $x_{1}, x_{2} \in R$ and $x_{1}>x_{2}$ $\Rightarrow a x_{1}<a x_{2}$ for some $a>0$ $\Rightarrow a x_{1}+b<a x_{2}+b$ for some...

Solution: It is given that, $f(x)=a x+b, a>0$ Suppose $x_{1}, x_{2} \in R$ and $x_{1}>x_{2}$ $\Rightarrow a x_{1}>a x_{2}$ for some $a>0$ $\Rightarrow a x_{1}+b>a x_{2}+b$ for some...

Solution: Case I When $a>1$ Suppose $\mathrm{x}_{1}, \mathrm{x}_{2} \in(0, \infty)$ We have, $\mathrm{x}_{1}<\mathrm{x}_{2}$ $\begin{array}{l} \Rightarrow \log _{e} x_{1}<\log _{e} x_{2} \\...

Solution: Suppose $x_{1}, x_{2} \in(0, \infty)$ We have, $x_{1}<x_{2}$ $\begin{array}{l} \Rightarrow \log _{\mathrm{e}} \mathrm{x}_{1}<\log _{\mathrm{e}} \mathrm{x}_{2} \\ \Rightarrow...