Question

Find the derivative of e^6x.

e^6x
6xe^6x
6e^6x
6e^x

For f(x) = e2sin(x) use your graphing calculator to find the number of zeros for f '(x) on the closed interval [0, 2π].

3
4
1
2

If f(x) = 5x4 tan−1x, find f '(x).

Answer 1

$\frac{d}{dx} e^{6x} =e^{6x} \, \frac{d}{dx} (6x) = 6 e^{6x}$
Answer: $6 e^{6x}$

$f(x)=e^{2sin(x)}$$f'(x) = e^{2\,sin(x)} \, 2 \, cos(x)=2\,cos(x) \, e^{2\,sin(x)}$
From the calculator, the zeros of f'(x) on the closed interval [0, 2π] are
$x= \frac{ \pi }{2} , \,\, \frac{3 \pi }{2}$
Answer: 2 zeros

f(x) = 5x⁴ tan⁻¹x
Note that
$\frac{d}{dx} tan^{-1}x = \frac{1}{1+x^{2}}$
Therefore
$f'(x) = 20x^{3} \, tan^{-1}x + 5x^{4} ( \frac{1}{1+x^{2}} ) \\ f'(x) = 5x^{3}(4\,tan^{-1}x + \frac{x}{1+x^{2}} )$
Answer: $5x^{3}(4\,tan^{-1}x + \frac{x}{1+x^{2}} )$