Question

Find the derivative of the function

f(x)=(2x+1)ln(2x+7)
and then evaluate it at x=0 .

f′(0)=

Answer 1

`f'(x)=2\ln(2x+7) + \left((2x+1)/(2x+7)\right)`

`f'(0) = 2ln(7) + (1)/(7)`

Explanation:

Recall that the derivative of a product of two functions
`f(x)=g(x)h(x)` is defined as

`(df(x))/(dx)=f'(x)= (dg(x))/(dx)h(x) + g(x)(dh(x))/(dx)`

Let
`g(x)=2x+1` and
`h(x)=\ln(2x+7)` so f'(x) is

`f'(x)=2\ln(2x+7) + \left((2x+1)/(2x+7)\right)`

Evaluated at x = 0, the value of the derivative is

`f'(0) = 2ln(7) + (1)/(7)`