Question

What is the derivative of `(1-tanx)^2`?

Answer 1

The power rule states that `D(f^n(x))=nf^{n-1}(x)f'(x)`.
So, in particular, `Df^2(x)=2f(x)f'(x)`.

In your case, `f(x)=1-tan(x)`. The only thing to do is to calculate `f'(x)`. The derivative of a sum is the sum of the derivatives, and the derivative of a constant is zero. So, you have that
`f'(x)=-Dtan(x)`.

And since `Dtan(x)=1/cos^2(x)`, sticking the pieces together gives
`Df^2(x)=2f(x)f'(x)=2(1-tan(x))/cos^2(x)`