How do you find the derivative of f(x)=ln (x^2+2)?

2 Answers
Aug 24, 2016

Since this is a composite function, you need to use the chain rule

Explanation:

The chain rune says that the derivative of a composition of functions f(x)=g(h(x)) is:

f'(x)=g'(h(x)) * h'(x). But:

g'(h(x))=ln'(x^2+2)=1/(x^2+2), and

h'(x)=2x, so the full derivative is:

f'(x)=1/(x^2+2)*2x=(2x)/(x^2+2

Aug 24, 2016

f'(x)=(2x)/(x^2+2).

Explanation:

Let y=f(x)=ln(x^2+2)

:. e^y=x^2+2.............(star).

rArr d/dx(e^y)=d/dx(x^2+2)

But, by the Chain Rule, d/dx(e^y)=d/dy(e^y)*dy/dx=e^y*dy/dx, and,

d/dx(x^2+2)=d/dx(x^2)+d/dx(2)=2x+0=2x.

Hence, e^y*dy/dx=2x

:. f'(x)=dy/dx=(2x)/e^y=(2x)/(x^2+2).

Enjoy Maths.!