Calculus trig differentiation of $f(w) = 4\sin^{2}( \frac{w}{2})-\sin^{2}(w)$

Question

$$ \begin{align} f(w) =& 4\sin^{2}\left( \frac{w}{2}\right)-\sin^{2}(w) \\ f’(w) =& 8\sin\left(\frac{w}{2}\right)\cos\left(\frac{w}{2}\right)-2\sin w \cos w \end{align} $$

Have I differentiated correctly?

The correct answer is

$ f’(w) = 2\sin(w)(1-\cos(w))$

Answer 1

You did a good job but forgot to differentiate $w/2$ $$f’(w) = 8\sin(w/2)\cos(w/2)\times \color{red}{\frac 12}-2\sin w \cos w$$

$$f’(w) = 4\sin(w/2)\cos(w/2)-2\sin w \cos w$$ $$f’(w) = 2\sin(w)-2\sin(w) \cos(w)$$ $$f’(w) = 2\sin(w)(1-\cos(w))$$

Answer 2

In the first part you differentiated correctly w.r.t. $w/2$ which when differentiated again w.r.t. $w$ in a chain rule needed to be to multiplied with $\dfrac12$.. that you have missed. All else is fine.