if ( f(x)=sin ^{2}(x) ), find ( f^{prime}(x)= ). find ( f^{prime}(4)= ). note: you can earn partial credit…

Explanation:

Step1: Apply the chain rule

Let (u = \sin(x)), then (f(x)=u^{2}). The chain rule states that (\frac{df}{dx}=\frac{df}{du}\cdot\frac{du}{dx}). For (f = u^{2}), (\frac{df}{du}=2u). For (u=\sin(x)), (\frac{du}{dx}=\cos(x)). So (f^{\prime}(x)=2\sin(x)\cos(x)).

Step2: Simplify (f^{\prime}(x))

Using the double - angle formula (\sin(2\alpha)=2\sin\alpha\cos\alpha), we can rewrite (f^{\prime}(x)) as (f^{\prime}(x)=\sin(2x)).

Step3: Find (f^{\prime}(4))

Substitute (x = 4) into (f^{\prime}(x)=\sin(2x)). (f^{\prime}(4)=\sin(8)).

Answer:

(f^{\prime}(x)=\sin(2x)) (f^{\prime}(4)=\sin(8))