assignment 5: problem 7\n(1 point)\nif ( f(x)=5 x sin (x) cos (x) ), find ( f^{prime}(x) ).\n(…

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 is ((uv)^\prime = u^\prime v+uv^\prime), where (u = \frac{5}{2}x) and (v=\sin(2x)) (u^\prime=\frac{5}{2}), (v^\prime = 2\cos(2x)) (f^\prime(x)=\frac{5}{2}\sin(2x)+\frac{5}{2}x\times2\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))