Question

How do you differentiate `f(x)=sqrt(1-sin^2x)`?

Answer 1

Please see the explanation.

Explanation:

My answer is incorrect please see Jim's answer

Answer 2

Please see below.

Explanation:

`f(x) = sqrt(1-sin^2x) = sqrt(cos^2x) = abs(cosx) = {(cosx,"if", cosx >=0),(-cosx,"if",cosx < 0):}`

So

`f(x) = {(cosx,"if",-pi/2+pik <= x <= pi/2+pik),(-cosx,"if",pi/2+pik < x < (3pi)/2+pik) :}` where `k` is an integer.

Therefore,

`f'(x) = {(-sinx,"if",-pi/2+pik < x < pi/2+pik),(sinx,"if",pi/2+pik < x < (3pi)/2+pik) :}` where `k` is an integer.

`f` is not differentiable at `x=pi/2+pik`

Here is the graph of `f`

graph{abs(cosx) [-5.28, 5.82, -1.885, 3.664]}