What is the derivative of arctan(6x)?

1 Answer
Aug 10, 2015

6/(1+36x^2)

Explanation:

Recap that d/dx arctan (x) = 1/(1 + x^2)

By the chain rule, if y is a function of u and u is a function of x, then dy/dx = dy/(du) * (du)/dx

Let u=6x \Rightarrow (du)/(dx) = 6
y=arctan(6x)=arctan(u) \Rightarrow dy/(du) = 1/(1+u^2)

Therefore by the chain rule,
dy/dx = dy/(du) * (du)/dx
d/dx (y) = 1/(1+u^2) * 6

Re-substituting u=6x and y=arctan(u)=arctan(6x):
d/dx arctan(6x) = 1/(1+(6x)^2) * 6 = 6/(1+36x^2)