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