How do you differentiate # f(x) = tan^2(3/x) #?

1 Answer
Feb 2, 2016

#f'(x)=(-6sec^2(3/x)tan(3/x))/x^2#

Explanation:

The first issue is the squared function. This will require the chain rule. Treat the problem like you would if it were #x^2#, which gives a derivative of #2x#, except that this must be multiplied by the derivative of the function that was squared as well.

#f'(x)=2tan(3/x)*d/dx[tan(3/x)]#

To differentiate the tangent function, use the chain rule again. Recall that the derivative of #tanx# is #sec^2x#.

#f'(x)=2tan(3/x)sec^2(3/x)*d/dx[3/x]#

To differentiate #3/x#, rewrite it as #3x^-1# and then use the power rule (no chain rule needed here).

#f'(x)=2tan(3/x)sec^2(3/x)(-3x^-2)#

This can be rewritten as

#f'(x)=(-6sec^2(3/x)tan(3/x))/x^2#