Question

exercises 3.5 derivatives of trigonometric functions
score: 18/37 answered: 10/18
question 11
textbook videos +
if ( f(x)=5 x sin x cos x ), find
( f^{prime}(x)= )
find ( f^{prime}(4)= )
question help: video message instructor
submit question jump to answer

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:

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