What's the derivative of #f(x) = (1/3) (arctan(3x))^2#?

1 Answer
mason m
Jan 31, 2016

#f'(x)=(2arctan(3x))/(1+9x^2)#

Explanation:

The first issue is the squared function. Use the chain rule:

#f'(x)=2/3(arctan(3x))^1d/dx(arctan(3x))#

To differentiate the #arctan# function, use the chain rule again. The general rule for #arctan# functions is #d/dx(arctan(u))=(u')/(1+u^2)#. Here, #u=3x#, so

#f'(x)=2/3arctan(3x)*3/(1+9x^2)#

This simplifies to be

#f'(x)=(2arctan(3x))/(1+9x^2)#

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