If $$ f(x)=\sin ^{2} x $$, find $$ f^{\prime}(x) $$ Find $$ f^{\prime}(4) $$

Question

UpStudy AI · Accepted Answer

To differentiate f(x) = sin²x, we use the chain rule. Let u = sin x, so that f(x) = u². Then

f'(x) = 2u · (du/dx) = 2 sin x · cos x.

This expression can be simplified using the double-angle formula for sine (sin 2x = 2 sin x cos x). Thus, we have

f'(x) = sin 2x.

To find f'(4), substitute x = 4 into the derivative:

f'(4) = sin(2 · 4) = sin 8.

So, f'(4) = sin 8 (with x given in radians).
$$ f^{\prime}(x) = \sin 2x $$ $$ f^{\prime}(4) = \sin 8 $$

If \( f(x)=\sin ^{2} x \), find \( f^{\prime}(x) \) Find \( f^{\prime}(4) \)