Question

What is the second derivative of f(x)=ln (x+3)?

Answer 1

`f''(x) = -1/(x+3)^2`

Explanation:

NOTE that the Derivative of `ln(x) = 1/x` and derivative of `1/x = -1/x^2`

`f'(x)=(d{ln(x+3)})/dx =1/(x+3)`

`f''(x) = (d{1/(x+3)})/dx = -1/(x+3)^2`

Answer 2

`f''(x)=-1/(x+3)^2`

Explanation:

`"differentiate using the "color(blue)"chain rule"`

`"Given "y=f(g(x))" then"`

`dy/dx=f'(g(x))xxg'(x)larrcolor(blue)"chain rule"`

`rArrf'(x)=1/(x+3)xxd/dx(x+3)=1/(x+3)=(x+3)^-1`

`"differentiate "f'(x)" using the chain rule"`

`rArrf''(x)=-(x+3)^-2xxd/dx(x+3)=-1/(x+3)^2`