Practice Problems

1. Differentiate

$$\frac{d}{dx}\, e^{\cos(3x)}$$

2. Differentiate

$$\frac{d}{dx}\,\sin(x^2+1)$$

3. Differentiate

$$\frac{d}{dx}\,\big(\cos(x^3)\big)^2$$

4. Differentiate

$$\frac{d}{dx}\,\ln(5x^2+1)$$

5. Differentiate

$$\frac{d}{dx}\,\sqrt{\,1+3x^4\,}$$

6. Differentiate

$$\frac{d}{dx}\,(2x+1)^{10}$$

7. Differentiate

$$\frac{d}{dx}\,\arctan\!\big(\sqrt{\,1-x^2\,}\big) \qquad(\text{assume }|x|<1)$$

8. Differentiate

$$\frac{d}{dx}\,\log_a(3x+5) \quad (a>0,\ a\neq 1)$$

9. Differentiate

$$\frac{d}{dx}{e}^{x^2\sin x}$$

$$\frac{d}{dx}\,e^{\,x^2\sin x}$$

10. Differentiate (away from points where $\sin(2x)=0$)

$$\frac{d}{dx}\,\big|\sin(2x)\big|$$

11. If $f$ is differentiable and $a,b$ are constants, find

$$\frac{d}{dx}\,f(ax^2+b)$$

Solutions

1)

$$\frac{d}{dx}\, e^{\cos(3x)} = e^{\cos(3x)}\cdot \frac{d}{dx}\big(\cos(3x)\big) = e^{\cos(3x)}\cdot\big(-\sin(3x)\cdot 3\big) = -3\sin(3x)\,e^{\cos(3x)}.$$

2)

$$\frac{d}{dx}\,\sin(x^2+1) = \cos(x^2+1)\cdot \frac{d}{dx}(x^2+1) = 2x\,\cos(x^2+1).$$

3) Write $u=\cos(x^3)$. Then $\frac{d}{dx}(u^2)=2u\,u’$.

$$\frac{d}{dx}\,\big(\cos(x^3)\big)^2 = 2\cos(x^3)\cdot\big(-\sin(x^3)\cdot 3x^2\big) = -6x^2\sin(x^3)\cos(x^3).$$

4)

$$\frac{d}{dx}\,\ln(5x^2+1) = \frac{1}{5x^2+1}\cdot (10x) = \frac{10x}{5x^2+1}.$$

5) Write $\sqrt{\,1+3x^4\,}=(1+3x^4)^{1/2}$.

$$\frac{d}{dx}\,(1+3x^4)^{1/2} = \tfrac12(1+3x^4)^{-1/2}\cdot 12x^3 = \frac{6x^3}{\sqrt{\,1+3x^4\,}}.$$

6) Power rule with inner derivative:

$$\frac{d}{dx}\,(2x+1)^{10} = 10(2x+1)^9\cdot 2 = 20(2x+1)^9.$$

7) Let $u=\sqrt{1-x^2}$. Then $\dfrac{d}{dx}\arctan(u)=\dfrac{u’}{1+u^2}$.
Here $u’=\dfrac{-x}{\sqrt{1-x^2}}$ and $1+u^2=1+(1-x^2)=2-x^2$.

$$\frac{d}{dx}\,\arctan\!\big(\sqrt{1-x^2}\big) = \frac{-x/\sqrt{1-x^2}}{\,2-x^2\,} = \frac{-x}{(2-x^2)\sqrt{1-x^2}} \quad(|x|<1).$$

8) $\dfrac{d}{dx}\log_a(3x+5)=\dfrac{1}{(3x+5)\ln a}\cdot 3$.

$$\frac{d}{dx}\,\log_a(3x+5)=\frac{3}{(3x+5)\ln a}.$$

9) Let $u=x^2\sin x$. Then $\dfrac{d}{dx}e^{u}=e^{u}u’$.
Product rule for $u$: $u’=2x\sin x+x^2\cos x$.

$$\frac{d}{dx}\,e^{x^2\sin x} = e^{x^2\sin x}\big(2x\sin x+x^2\cos x\big).$$

10) For $u(x)=\sin(2x)$ and $u(x)\neq 0$, $\dfrac{d}{dx}|u|=\dfrac{u}{|u|}\,u’$.
Here $u’ = 2\cos(2x)$.

$$\frac{d}{dx}\,|\sin(2x)| = \frac{\sin(2x)}{|\sin(2x)|}\cdot 2\cos(2x), \quad \text{undefined where }\sin(2x)=0.$$

11) Let $u=ax^2+b$. Then

$$\frac{d}{dx}\,f(ax^2+b)=f'(u)\cdot u’=f'(ax^2+b)\cdot (2ax).$$

When in doubt, name the inside $u(\,x\,)$ and differentiate in two beats: outer derivative at $u$, then multiply by $u’$. That simple habit saves a lot of mistakes.