Free Printable Chain Rule, Implicit, Logs and Exponential Derivatives

1.

Differentiate

y=3^{\left(x^2-1\right)}

a)

$x^2\left(\ln3\right)\left(3^{x^2-1}\right)$

b)

$\left(\ln3\right)\left(3^{x^2-1}\right)$

c)

$\left(2x\right)\left(3^{x^2-1}\right)$

d)

$2x\left(\ln3\right)\left(3^{x^2-1}\right)$

2.

Find $\frac{\text{d}y}{\text{d}x}$ if $y=e^{\sin\sqrt{x}}$

a)

$\left(\cos\sqrt{x}\right)e^{\sin\sqrt{x}}$

b)

$\frac{e^{\cos\sqrt{x}}}{2\sqrt{x}}$

c)

$\frac{\cos\sqrt{x}}{2\sqrt{x}}\left(e^{\sin\sqrt{x}}\right)$

d)

$\left(\sin\sqrt{x}\right)e^{\left(\sin\sqrt{x}-1\right)}$

3.

$f\left(x\right)=\sqrt{5+e^{2x}}$

Find f'(x) for

a)

$\frac{1}{2\sqrt{2e^{2x}}}$

b)

$\frac{e^{2x}}{\sqrt{5+e^{2x}}}$

c)

$\frac{5+e^{2x}}{\sqrt{5+e^{2x}}}$

d)

$\frac{xe^{\left(2x-1\right)}}{\sqrt{5+e^{2x}}}$

4.

If $y=\tan^{-1}\left(\cos\ x\right)$ , then $\frac{\text{d}y}{\text{d}x}$ =

a)

$\left(\sec^{-1}\left(\cos\ x\right)\right)^2$

b)

$\frac{1}{\left(\cos^{-1}x\right)^2+1}$

c)

$-\left(\sec^{-1}\left(\cos x\right)\right)^2\sin x$

d)

$\frac{-\sin x}{1+\cos^2x}$

5.

Let $f\left(x\right)=x^4+3x-2$ , and let g(x) denote the inverse of f. Then g'(2) is equal to:

a)

1/7

b)

1/35

c)

7

d)

35

6.

$g\left(x\right)=\sin^3\left(4x\right)$

Find g'(x) if

a)

$\cos^3\left(4x\right)$

b)

$3\sin^2\left(4x\right)\cos\left(4x\right)$

c)

$12\sin^2\left(4x\right)\cos\left(4x\right)$

d)

$4\cos^3\left(4x\right)$

7.

For what values of x does $y^2-2x^3-12x^2=0$ have horizontal tangent lines?

a)

x = 0 only

b)

x = 0 and x = - 4

c)

x = - 4 only

d)

x = - 4, x = 0, and x = 4

8.

If $f\left(x\right)=\left(x+1\right)\left(x^2-2\right)^3$ then f'(x)=

a)

$6x\left(x^2-2\right)^2$

b)

$6x\left(x+1\right)\left(x^2-2\right)^2$

c)

$3\left(x+1\right)\left(x^2-2\right)^2$

d)

$\left(x^2-2\right)^2\left(7x^2+6x-2\right)$

9.

Find the second derivative give $g\left(x\right)=\sin\ x^2$

a)

$2\left(\cos\ x^2-x\sin x^2\right)$

b)

$2\left(\cos x^2-2x^2\sin x^2\right)$

c)

$2x\cos x^2$

d)

$-4x\sin x^2$

10.

If $h\left(x\right)=\ln\left(2-3x\right)^5$ , find h'(0)

a)

5/2

b)

-2/3

c)

-5/3

d)

-15/2

Chain Rule, Implicit, Logs and Exponential Derivatives

Total questions: 10

Worksheet time: 19mins

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