Question

assignment 5: problem 7
(1 point)
if ( f(x)=5 x sin (x) cos (x) ), find ( f^{prime}(x) ).
( f^{prime}(x)=)
find ( f^{prime}(4) ).
( f^{prime}(4)=)

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:

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