# Unit 02: Differentiation

Here is the list of important questions.

• Differentiate $\frac{(x^2+1)^2}{x^2-1}$ $w.r.t.x$. — BSIC Gujranwala (2016)
• If $x=at^2$, $y=2at$. Find $\frac{dy}{dx}$ — BSIC Gujranwala (2016)
• Differentiate $x^2-\frac{1}{x^2}$ $w.r.t.x^2$. — BSIC Gujranwala (2016)
• Prove that $\frac{d}{dx}(tan^{-1}x)=\frac{1}{1+x^2}$ — BSIC Gujranwala (2016)
• Prove that $\frac{d}{dx}(sinh^{-1}x)=\frac{1}{\sqrt{1+x^2}}$ — BSIC Gujranwala (2016)
• If $y=x^2ln(\frac{1}{x})$. Find $\frac{dy}{dx}$. — BSIC Gujranwala (2016)
• If $x=sin\theta$, $y=sin m\theta$. Find $\frac{dy}{dx}$. — BSIC Gujranwala (2016)
• Apply Maclaurin series to expand $cosx =1-\frac{x^2}{2!}$ — BSIC Gujranwala (2016)
• Differentiate $cos^2x$ $w.r.t.sin^2x$. — BSIC Gujranwala (2016)
• Using differential find $\frac{dy}{dx}$, When $x^2+2y^2=16$ — BSIC Gujranwala (2016)
• Show that $\frac{dy}{dx}=\frac{y}{x}$ if $\frac{y}{x}=tan^{-1}(\frac{x}{y})$ — BSIC Gujranwala (2016)
• Find $\frac{dy}{dx}$, if $y^2-xy-x^2+4=0$ — BSIC Gujranwala (2015)
• Prove that $\frac{dy}{dx}({log_a}^x)=\frac{1}{x \ln a}$ — BSIC Gujranwala (2015)
• Find $\frac{dy}{dx}$, if $y=(\ln x)^{\ln x}$ — BSIC Gujranwala (2015)
• Find $y_4$ if $y=\cos ^3x$ — BSIC Gujranwala (2015)
• Expand $a^x$ in Meclaurin series. — BSIC Gujranwala (2015)
• Prove that derivative of a constant is zero. — BSIC Gujranwala (2015)
• Determine the interval in which $f$ is increasing if $f(x)=x^3-6x^2+9x$. — BSIC Gujranwala (2015)
• Differentiate $\cos \sqrt{x}+\sqrt{\sin x}$ with respect to $x$. — BSIC Gujranwala (2015)
• Use differentials to approximate the value of $(31)^{\frac{1}{5}}$ — BSIC Gujranwala (2015)
• If $y=x^4+2x^2+2$, prove that $\frac{dy}{dx}=40x\sqrt{y-1}$— BSIC Gujranwala (2015)
• Find $\frac{dy}{dx}$, if $x=\frac{a(1-t^2)}{1+t^2}$ and $y=\frac{2bt}{1+t^2}$. — BSIC Gujranwala (2015)
• Differentiate $\cos \sqrt{x}$ with respect to $x$ by ab-initio method. — BSIC Gujranwala (2015)
• A box with a square base and open top is to have avolume of $4$ cubic dm. Find the dimensions of the box which will require the least material? – BSIC Gujranwala (2015)
• Find the extreme values of the function $f(x)=\sin x +\cos x$ occurring in the intial $[0,2\pi]$ – BSIC Gujranwala (2015)
• Find $\frac{dy}{dx}$, if $x^2-4xy-5y^2=0$ — FBSIC (2016)
• If $y=\sqrt{\tan x+\sqrt{\tan x+\sqrt{x}}+\ldots}$, prove that $(2y-1)\frac{dy}{dx}=\sec ^2x$. — FBSIC (2016)
• Find $\frac{dy}{dx}$, if $y=x e^{\sin x}$. — FBSIC (2016)
• Show that $\frac{dy}{dx}=\frac{y}{x}$, if $\frac{y}{x}=\tan ^{-1}\frac{y}{x}$. — FBSIC (2016)
• Differentiate $(\sqrt{x}-\frac{1}{\sqrt{x}})$ w.r.t. $x$.— BSIC Rawalpandi (2017)
• Find $\frac{dy}{dx}$, if $y=\sqrt{x+\sqrt{x}}$.— BSIC Rawalpandi (2017)
• Differentiate $x^2 \sec 4x$ w.r.t. $x$..— BSIC Rawalpandi (2017)
• Find $\frac{dy}{dx}$, if $x=y \sin y$.— BSIC Rawalpandi (2017)
• Find $f'(x)$ if $f(x)=x^2 \ln \sqrt{x}$— BSIC Rawalpandi (2017)
• Find $y_2$ if $y-\cos ^3x$.— BSIC Rawalpandi (2017)
• Find $\frac{dy}{dx}$, if $y=x e^{\sin x}$.— BSIC Rawalpandi (2017)
• Apply maclaurins series expansions to prove that $e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots$— BSIC Rawalpandi (2017)
• Determine the intervals in which $f(x)=\cos x: x\in (-\frac{\pi}{2},\frac{\pi}{2})$ is increasing or decreasing function.— BSIC Rawalpandi (2017)
• If $x=\sin \theta, \gamma=\sin (m\theta)$, then prove that $(1-x^2)y_2-xy_1+m^2y+0$— BSIC Rawalpandi (2017)
• Using differential, find $\frac{dy}{dx}$ in the equation $x^2+2y^2=16$— BSIC Rawalpindi(2017)
• If $f(x)=x^2$, then find $f'(x)$ by defination. — BSIC Sargodha(2016)
• Differentiate $\frac{a+x}{a-x}$ w.r.t.$x$.— BSIC Sargodha(2016)
• If $x=\theta +\frac{1}{\theta}$ and $y=\theta +1$ then find $\frac{dy}{dx}$. — BSIC Sargodha(2016)
• find $\frac{dy}{dx}$ if $y=x \cos y$ — BSIC Sargodha(2016)
• If $y=e^{x^2+1}$ then find $\frac{dy}{dx}$. — BSIC Sargodha(2016)
• Find $f'(x)$, if $f(x)=\ln (e^x+e^{-x})$. — BSIC Sargodha(2016)
• If $y=\cos (ax+b)$ then find $y_1$. — BSIC Sargodha(2016)
• By maclaurins series, prove that $e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots$. — BSIC Sargodha(2016)
• Defined increasing and decreasing function. — BSIC Sargodha(2016)
• Prove that $y \frac{dy}{dx}+x=0$ if $x=\frac{1-t^2}{1+t^2}$, $y==\frac{2t}{1+t^2}$ . — BSIC Sargodha(2016)
• Define the derivative w.r.t.$x$. — BSIC Sargodha(2017)
• Differentiate w.r.t.$x$ $(x-5)(3-x)$ — BSIC Sargodha(2017)
• Differentiate w.r.t.$x$ $\frac{1}{a}\sin ^{-1}\frac{a}{x}$ — BSIC Sargodha(2017)
• Find $\frac{dy}{dx}$, if $y=x^2 \ln \sqrt{x}$ — BSIC Sargodha(2017)
• Find $\frac{dy}{dx}$, if $y=xe^{\sin x}$ — BSIC Sargodha(2017)
• Find $y_2$ if $y=(2x+5)^{\frac{1}{2}}$ — BSIC Sargodha(2017)
• What is the decreasing function. — BSIC Sargodha(2017)
• Find $\frac{dy}{dx}$, if $y= \ln \sqrt{\frac{x^2-1}{x^2+1}}$ — BSIC Sargodha(2017)
• Differentiate $\cos \sqrt{x}$ w.r.t.$x$ from first principle. — BSIC Sargodha(2017)
• fsc-part2-ptb/important-questions/unit-02-differentiation