Question

If f(x)=(4x^2tan(x))/(sec(x), how do I find f'(x) and f'(3)?

I tried finding f'(x) first, but my answer of (8xsec^2(x)(sec(x))-(4x^2tan(x)(sec(x)tan(x))))/(secx)^2 is incorrect. Do you have to do something other than the quotient rule or did I just do my math wrong?

Answer 1

`f'(x) = 8xsinx + 4x^2cosx`

Explanation:

We can rewrite as

`f(x) = 4x^2tanx(1/secx)`

`f(x) = 4x^2tanx(1/(1/cosx))`

`f(x) = 4x^2tanxcosx`

`f(x) = 4x^2sinx`

Now we can easily differentiate using the product rule.

`f'(x) = 8x(sinx) + cosx(4x^2)`

`f'(x) = 8xsinx + 4x^2cosx`

I'll let you evaluate `f'(3)`.

Hopefully this helps!