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

Question

10625 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-08-24T16:03:55

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

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#

Answer 2 · 2016-08-24T16:46:31

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

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.!