Question

Question #9043b

Answer 1

Use `ln(u/v) = lnu-lnv` to simplify the process.

Explanation:

(If `c` is a constant, then the whole thing is constant and the derivative is `0`)

I will assume that `c` is the independent variable. If we are differentiating with respect to some other variable, then we must apply the chain rule and multiply by the derivative of `c`.

`f(c) = ln((1+sinc)/(1-sinc)) = ln(1+sinc)-ln(1-sinc)`

Now use `d/(dc) (lnu) = 1/u (du)/(dc)` (chain rule) to get

`f'(c) = 1/(1+sinc) * (cosc) - 1/(1-sinc) (-cosc)`

`= cosc/(1+sinc) + cosc/(1-sinc)`

`= (2cosc)/(1-sin^2c)`

`= 2/cosc = 2sec c`.