Question

How do you differentiate `f(x)=(tanx-1)/secx` at `x=pi/3`?

Answer 1

`(1 + sqrt(3))/2`

Explanation:

Rewrite in sine and cosine.

`f(x) = (sinx/cosx- 1)/(1/cosx)`

`f(x) = ((sinx - cosx)/cosx)/(1/cosx)`

`f(x) = sinx - cosx`

We differentiate this using `d/dx(sinx) = cosx` and `d/dx(cosx) = -sinx`.

`f'(x) = cosx - (-sinx)`

`f'(x) = cosx + sinx`

We now evaluate `f'(pi/3)`:

`f'(pi/3) = cos(pi/3) + sin(pi/3)`

`f'(pi/3) = 1/2 + sqrt(3)/2`

`f'(pi/3) = (1 + sqrt(3))/2`

Hopefully this helps!