Question

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

Answer 1

`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`