What is the derivative of #arctan(6x)#?

1 Answer
Joel Kindiak
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)#

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