Question

find the derivative of $f(x)$. $f(x)=\ln(x^{2}+4)$ $f(x)=$

Explanation:

Step1: Use the chain rule

Let \(u = x^{2}+4\), then \(f(x)=\ln(u)\). The derivative of \(\ln(u)\) with respect to \(u\) is \(\frac{1}{u}\), and the derivative of \(u = x^{2}+4\) with respect to \(x\) is \(2x\).

Step2: Apply the chain rule formula

By the chain rule \((f(g(x)))^\prime=f^\prime(g(x))\cdot g^\prime(x)\), we have \(f^\prime(x)=\frac{1}{u}\cdot2x\).

Step3: Substitute \(u\) back

Since \(u = x^{2}+4\), then \(f^\prime(x)=\frac{2x}{x^{2}+4}\).

Answer:

\(\frac{2x}{x^{2}+4}\)