exercises 3.5 derivatives of trigonometric functions\nscore: 18/37 answered: 10/18\nquestion 11\ntextbook…

Explanation:

Step1: Simplify the function

Use the double - angle formula (\sin(2x)=2\sin x\cos x), so (f(x)=\frac{5}{2}x\sin(2x))

Step2: Apply the product rule

The product rule ((uv)^\prime = u^\prime v+uv^\prime), where (u = \frac{5}{2}x) and (v=\sin(2x)) (u^\prime=\frac{5}{2}), and (v^\prime = 2\cos(2x)) (using the chain rule ((\sin(u))^\prime=\cos(u)\cdot u^\prime) with (u = 2x)) (f^\prime(x)=\frac{5}{2}\sin(2x)+\frac{5}{2}x\cdot2\cos(2x)) (f^\prime(x)=\frac{5}{2}\sin(2x)+5x\cos(2x))

Step3: Calculate (f^\prime(4))

Substitute (x = 4) into (f^\prime(x)) (f^\prime(4)=\frac{5}{2}\sin(8)+5\times4\cos(8)) (f^\prime(4)=\frac{5}{2}\sin(8)+20\cos(8))

Answer:

(f^\prime(x)=\frac{5}{2}\sin(2x)+5x\cos(2x)) (f^\prime(4)=\frac{5}{2}\sin(8)+20\cos(8))