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

`(d)/(dx) e^(6x) =e^(6x) \, (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= ( \pi )/(2) , \,\, (3 \pi )/(2)`
Answer: 2 zeros

f(x) = 5x⁴ tan⁻¹x
Note that

`(d)/(dx) tan^(-1)x = (1)/(1+x^(2))`
Therefore

`f'(x) = 20x^(3) \, tan^(-1)x + 5x^(4) ( (1)/(1+x^(2)) ) \\ f'(x) = 5x^(3)(4\,tan^(-1)x + (x)/(1+x^(2)) )`
Answer:
`5x^(3)(4\,tan^(-1)x + (x)/(1+x^(2)) )`