calculus antiderivative/ derivative question

Question

find f if f'(x)=2/x, x<0 and f (-1)=2

Answer 1

f'(x) = 2/x = 2 (1/x)

To solve for f(x), we take the integral of f'(x), with the integral of 1/x being ln |x|:

f(x) = 2 ln |x| + C

Using the initial condition f(-1)=2, plug in -1 for x and set equal to 2:

f(-1) = 2 ln |-1| + C = 2

Solve for C:

C = 2 - 2 ln |-1| = 2 - 2 ln(1) = 2 - 2(0) = 2

Plug C into the equation for f(x):

f(x) = 2 ln |x| + 2