Question

find f(x).
f(x) = 4x^5 ln x
f(x) = \square

Explanation:

Step1: Apply product rule

The product rule states that if \(y = u\cdot v\), then \(y^\prime=u^\prime v + uv^\prime\). Let \(u = 4x^{5}\) and \(v=\ln x\). First, find \(u^\prime\) and \(v^\prime\).
For \(u = 4x^{5}\), using the power rule \((x^{n})^\prime=nx^{n - 1}\), we have \(u^\prime=4\times5x^{4}=20x^{4}\).
For \(v=\ln x\), the derivative \(v^\prime=\frac{1}{x}\).

Step2: Substitute into product rule

\(f^\prime(x)=u^\prime v+uv^\prime\)
\(f^\prime(x)=20x^{4}\ln x+4x^{5}\times\frac{1}{x}\)

Step3: Simplify the expression

\(4x^{5}\times\frac{1}{x}=4x^{4}\)
So \(f^\prime(x)=20x^{4}\ln x + 4x^{4}=4x^{4}(5\ln x + 1)\)

Answer:

\(4x^{4}(5\ln x + 1)\)