Problem Set - Differentiation

## The chain rule

 Let $g$ be a function. Show that $\frac{d{g}^{0}}{dx}=0\frac{dg}{dx}$. Let $g$ be a function. Show that $\frac{d{g}^{1}}{dx}=1{g}^{0}\frac{dg}{dx}$. Let $g$ be a function. Show that $\frac{d{g}^{2}}{dx}=2{g}^{1}\frac{dg}{dx}$. Let $g$ be a function. Show that $\frac{d{g}^{3}}{dx}=3{g}^{2}\frac{dg}{dx}$. Let $g$ be a function. Show that $\frac{d{g}^{4}}{dx}=4{g}^{3}\frac{dg}{dx}$. Let $g$ be a function. Show that $\frac{d{g}^{5}}{dx}=5{g}^{4}\frac{dg}{dx}$. Let $g$ be a function. Show that $\frac{d{g}^{n}}{dx}=n{g}^{n-1}\frac{dg}{dx}$. Let $f\left(y\right)=4{y}^{3}+7{y}^{2}+2y-13$ and let $g$ be a function. Show that $\frac{d\left(f\left(g\right)\right)}{dx}=\left(12{g}^{2}+14g+2\right)\frac{dg}{dx}\text{.}$ Let $f$ be a polynomial and let $g$ be a function. Show that $\frac{d\left(f\left(g\right)\right)}{dx}=\frac{df}{dg}·\frac{dg}{dx}$.

## Derivatives of the basic functions

 Explain why $\frac{d{e}^{x}}{dx}={e}^{x}$. Explain why $\frac{d\mathrm{sin}x}{dx}=\mathrm{cos}x$. Explain why $\frac{d\mathrm{cos}x}{dx}=-\mathrm{sin}x$. Explain why $\frac{d\mathrm{tan}x}{dx}={\mathrm{sec}}^{2}x$. Explain why $\frac{d\mathrm{cot}x}{dx}=-{\mathrm{cot}}^{2}x$. Explain why $\frac{d\mathrm{sec}x}{dx}=\mathrm{tan}x\mathrm{sec}x$. Explain why $\frac{d\mathrm{csc}x}{dx}=-\mathrm{cot}x\mathrm{csc}x$. Explain why $\frac{d\mathrm{ln}x}{dx}=\frac{1}{x}$. Explain why $\frac{d{\mathrm{sin}}^{-1}x}{dx}=\frac{1}{\sqrt{1-{x}^{2}}}$. Explain why $\frac{d{\mathrm{cos}}^{-1}x}{dx}=-\frac{1}{\sqrt{1-{x}^{2}}}$. Explain why $\frac{d{\mathrm{tan}}^{-1}}{dx}=\frac{1}{1+{x}^{2}}$. Explain why $\frac{d{\mathrm{cot}}^{-1}x}{dx}=-\frac{1}{1+{x}^{2}}$. Explain why $\frac{d{\mathrm{csc}}^{-1}x}{dx}=-\frac{1}{\left|x\right|\sqrt{{x}^{2}-1}}$.

## Computing some derivatives

 Find $\frac{dy}{dx}$ when $y=\left(2x+3\right)\left(5x+6\right)$. Find $\frac{dy}{dx}$ when $y=\left(x+\frac{1}{x}\right)\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)$. Find $\frac{dy}{dx}$ when $y={\left(2x-5\right)}^{2}{\left(3x-4\right)}^{3}$. Find $\frac{dy}{dx}$ when $y=\left(e{x}^{2}+\frac{\pi }{{x}^{3}}+{x}^{7/2}\right)$. Find $\frac{dy}{dx}$ when $y={\left(\frac{x-3}{x-4}\right)}^{2}$. Find $\frac{dy}{dx}$ when $y=\frac{3x+5}{4-{x}^{2}}$. Find $\frac{dy}{dx}$ when $y=\frac{x}{\sqrt{1-2x}}$. Find $\frac{dy}{dx}$ when $y=\frac{1+\sqrt{x}}{1-\sqrt{x}}$. Find $\frac{dy}{dx}$ when $y=\frac{2\left(x+1\right)}{{x}^{2}+2x-3}$. Find $\frac{dy}{dx}$ when $y=\frac{\sqrt{a+x}-\sqrt{a-x}}{\sqrt{a+x}+\sqrt{a-x}}$. Find $\frac{dy}{dx}$ when $y=\frac{{x}^{2}-2}{x-1}$. Find $\frac{dy}{dx}$ when $y=\frac{\sqrt{x}}{\sqrt{x-3}}$. Find $\frac{dy}{dx}$ when $y=\frac{{x}^{n}+1}{{x}^{n}-1}$. Find $\frac{dy}{dx}$ when $y=\frac{\sqrt{1+{x}^{2}}}{\sqrt{1-{x}^{2}}}$. Find $\frac{dy}{dx}$ when $y=\frac{2{x}^{2}-1}{x\sqrt{1+{x}^{2}}}$. Find $\frac{dy}{dx}$ when $y={u}^{n}$. Find $\frac{dy}{dx}$ when $y=\sqrt{1-{x}^{2}}$.

## Correcting derivative identities

 Correct the identity $\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}+u\frac{dv}{dx}}{{v}^{2}}$ Correct the identity $\frac{d}{dx}\left(u+v\right)=\frac{du}{dv}-\frac{dv}{dx}$ Correct the identity $\frac{d}{dx}\left(uv\right)=\frac{du}{dx}·\frac{dv}{dx}$

## Verifying derivative identities

 If $y={x}^{7/2}$ show that $2x\frac{dy}{dx}-7y=0$. If $y=3-{x}^{2}$ show that ${\left(\frac{dy}{dx}\right)}^{2}-4{x}^{2}=0$. If $y=\sqrt{x}+\frac{1}{\sqrt{x}}$ show that $2x\frac{dy}{dx}+y-\sqrt{x}=0$. If $y=1+x+\frac{{x}^{2}}{2!}+\frac{{x}^{3}}{3!}+\cdots +\frac{{x}^{n}}{n!}$ show that $\frac{dy}{dx}-y+\frac{{x}^{n}}{n!}=0$. If $y=1+x+\frac{{x}^{2}}{2!}+\frac{{x}^{3}}{3!}+\cdots$ show that $\frac{dy}{dx}-y=0$. If $z=\frac{3}{1+t}$ show that $3t\frac{dz}{dt}=z\left(z-3\right)$. If $y=\frac{1}{a-z}$ show that $\frac{dz}{dy}={\left(z-a\right)}^{2}$. If $y=\frac{x}{x-p}$ show that $x\frac{dy}{dx}=y\left(1-y\right)$. If $y=x-\sqrt{1+{x}^{2}}$ show that $\left(1+{x}^{2}\right){\left(\frac{dy}{dx}\right)}^{2}={y}^{2}$. If $y={x}^{2}$ show that ${\left(\frac{dy}{dx}\right)}^{2}=4y$. If $y=\sqrt{1+{x}^{5}}$ show that $\frac{dy}{dx}=\frac{5{x}^{4}}{2y}$.

## Derivatives at a point

 Find $\frac{dy}{dx}$ at $x=3$ when $y={x}^{6}+3{x}^{2}+5$. Find ${\frac{dy}{dx}|}_{x=3}$ when $y=\left(x+1\right)\left(x+2\right)$.

## Derivatives with respect to functions

 Differentiate ${t}^{2}-\frac{4}{{t}^{4}}$ with respect to ${t}^{5}$. Differentiate $\frac{{x}^{2}}{1+{x}^{2}}$ with respect to ${x}^{2}$. Differentiate $\frac{ax+b}{cx+d}$ with respect to $\frac{a\prime x+b\prime }{c\prime x+d\prime }$. Differentiate ${x}^{3}$ with respect to ${x}^{2}$. Differentiate $\frac{\sqrt{1+{x}^{2}}-\sqrt{1-{x}^{2}}}{\sqrt{1+{x}^{2}}+\sqrt{1-{x}^{2}}}$ with respect to $\sqrt{1-{x}^{4}}$. Differentiate $\frac{x}{1+{x}^{2}}$ with respect to ${x}^{3}$. Differentiate $x-\sqrt{1-{x}^{2}}$ with respect to $\sqrt{1-{x}^{2}}$. Differentiate $7{x}^{5}-11{x}^{2}$ with respect to $7{x}^{2}-15x$.

## Derivatives of parametric functions

 Find $\frac{dy}{dx}$ when $x=pt$ and $y=p/t$. Find $\frac{dy}{dx}$ when $x=a{t}^{2}$ and $y=2at$. Find $\frac{dy}{dx}$ when $y=\frac{2a{t}^{2}}{1+{t}^{2}}$ and $x=\frac{2a}{1+{t}^{2}}$. Find $\frac{dy}{dx}$ when $a\frac{1-{t}^{2}}{1+{t}^{2}}$ and $y=b\frac{2t}{1+{t}^{2}}$. Find $\frac{dy}{dx}$ when $x=a\sqrt{\frac{{t}^{2}-1}{{t}^{2}+1}}$ and $y=at\sqrt{\frac{{t}^{2}-1}{{t}^{2}+1}}$. Find $\frac{dy}{dx}$ when $x=a\frac{1-{t}^{2}}{1+{t}^{2}}$ and $y=\frac{2bt}{1-{t}^{2}}$. Find $\frac{dy}{dx}$ when $x=\frac{3at}{1+{t}^{3}}$ and $y=\frac{3a{t}^{2}}{1+{t}^{3}}$. Find $\frac{dy}{dx}$ when $x=\frac{1-{t}^{2}}{1+{t}^{2}}$ and $y=\frac{2t}{1+{t}^{2}}$.

## Implicit differentiation

 Find $\frac{dy}{dx}$ when ${x}^{4}+{y}^{4}=4{a}^{2}{x}^{2}{y}^{2}.$ Find $\frac{dy}{dx}$ when $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1.$ Find $\frac{dy}{dx}$ when ${x}^{5}+{y}^{5}-5a{x}^{2}{y}^{2}=0.$ If $a{x}^{2}+b{y}^{2}+2gx+2fy+2hxy+c=0$ show that $\frac{dy}{dx}+\frac{ax+hy+g}{hx+by+f}=0.$ If $xy+px+q=0$ prove that ${x}^{2}\frac{dy}{dx}$ is always constant. Find $\frac{dy}{dx}$ when $a{x}^{2}+2hxy+b{y}^{2}+2gx+2fy+c=0.$

## Derivatives with trigonometric functions.

 Find $\frac{dy}{dx}$ when $y=\mathrm{sin}\left(3x+2\right).$ Find $\frac{dy}{dx}$ when $y=\sqrt{\mathrm{sin}{x}^{4}}.$ Find $\frac{dy}{dx}$ when $y={x}^{2}\mathrm{sin}x.$ Find $\frac{dy}{dx}$ when $y=\mathrm{tan}x\mathrm{sin}2x.$ Find $\frac{dy}{dx}$ when $y=\mathrm{sin}{x}^{2}-\frac{\mathrm{tan}x}{1+{x}^{2}}.$ Find $\frac{dy}{dx}$ when $y=\frac{2\mathrm{cos}x-x}{x+2}.$ Find $\frac{dy}{dx}$ when $y=\left(1+{x}^{2}\right)+\frac{x}{\mathrm{sin}x}.$ Find $\frac{dy}{dx}$ when $y=\frac{\mathrm{sin}2x}{\mathrm{cos}x}.$ Find $\frac{dy}{dx}$ when $y=\mathrm{sin}\left(\frac{x}{3}\right)\mathrm{csc}\left(\frac{2x}{3}\right).$ Find $\frac{dy}{dx}$ when $y=\mathrm{sin}\left(\mathrm{sin}x+\mathrm{cos}x\right).$ Find $\frac{dy}{dx}$ when $y=\sqrt{{\mathrm{sec}}^{2}x+{\mathrm{csc}}^{2}x}.$ Find $\frac{dy}{dx}$ when $y=\left({x}^{2}-1\right)\left(\mathrm{cot}x+\frac{\mathrm{tan}x}{1+{x}^{2}}\right).$ Find $\frac{dy}{dx}$ when $y=\sqrt{\frac{\mathrm{cos}\theta -\mathrm{sin}\theta }{\mathrm{cos}\theta +\mathrm{sin}\theta }}.$ Find $\frac{dy}{dx}$ when $y=\frac{\mathrm{sec}x+\mathrm{tan}x}{\mathrm{sec}x-\mathrm{tan}x}.$ Find $\frac{dy}{dx}$ when $y=\sqrt{\frac{1-\mathrm{cos}x}{1+\mathrm{cos}x}}.$ Find $\frac{dy}{dx}$ when $y={x}^{3}{\mathrm{tan}}^{2}\left(\frac{x}{2}\right).$ If $y=\mathrm{tan}\left(\mathrm{cos}\left(\mathrm{sin}\theta \right)\right)$ find $\frac{dy}{dx}.$

## Derivatives with exponentials and logs.

 Find $\frac{dy}{dx}$ when $y=\left(e{x}^{2}+\frac{\pi }{{x}^{3}}+{x}^{\frac{7}{2}}\right).$ Find $\frac{dy}{dx}$ when $y={a}^{ax+b}.$ Find $\frac{dy}{dx}$ when $y={a}^{{x}^{3}}.$ Find $\frac{dy}{dx}$ when $y={6}^{2x}.$ Find $\frac{dy}{dx}$ when $y=\mathrm{ln}\left(a{x}^{2}+b\right).$ Find $\frac{dy}{dx}$ when $y={e}^{3\mathrm{ln}x}.$ Find $\frac{dy}{dx}$ when $y={e}^{2x}-{e}^{-2x}.$ Find $\frac{dy}{dx}$ when $y={e}^{{x}^{2}+2x}.$ Find $\frac{dy}{dx}$ when $y={a}^{x}{x}^{a}.$ Find $\frac{dy}{dx}$ when $y=x{e}^{x}.$