How do you differentiate f(x) = 1/sqrt(arctan(2x^3) f(x)=1arctan(2x3) using the chain rule?

1 Answer
Nov 5, 2016

f'(g(x))g'(x) = 1/2(arctan(2x^3))*(6x^2)/(1+(2x^3)^2)

Explanation:

first, take the derivative of the denominator using chain rule:
f=sqrt(x)
f'=1/2x^(-1/2)
g=arctan(2x^3)
g'=(6x^2)/(1+(2x^3)^2)

f'(g(x))g'(x) = 1/2(arctan(2x^3))^(-1/2)*(6x^2)/(1+(2x^3)^2)

that's it! if you want to simplify it further you can but my calc teacher only requires this.

note: recognize that the derivative of arctan has the chain rule built into it already if that makes sense, as d/dx[arctan(u)]=(u')/(1+u^2)