Question

let
$f(x)=\ln x^{6}(x + 2)^{5}(x^{2}+7)^{5}$
hint: use the properties of logarithms to expand the function before taking the derivative.
$f(x)=$
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Explanation:

Step1: Expand the function using logarithm properties

Using the property \(\ln(ab)=\ln a+\ln b\) and \(\ln(a^n) = n\ln a\), we have:
\(f(x)=\ln(x^{6})+\ln((x + 2)^{5})+\ln((x^{2}+7)^{5})\)
\(f(x)=6\ln x+5\ln(x + 2)+5\ln(x^{2}+7)\)

Step2: Differentiate term - by - term

The derivative of \(\ln u\) with respect to \(x\) is \(\frac{u^{\prime}}{u}\) (by the chain rule \((\ln u)^\prime=\frac{u^\prime}{u}\)).

• For \(y = 6\ln x\), \(y^\prime=\frac{6}{x}\) (since \(u = x\) and \(u^\prime=1\)).
• For \(y = 5\ln(x + 2)\), \(u=x + 2\), \(u^\prime=1\), so \(y^\prime=\frac{5}{x + 2}\).
• For \(y = 5\ln(x^{2}+7)\), \(u=x^{2}+7\), \(u^\prime = 2x\), so \(y^\prime=\frac{5\times2x}{x^{2}+7}=\frac{10x}{x^{2}+7}\).

Step3: Combine the derivatives

\(f^\prime(x)=\frac{6}{x}+\frac{5}{x + 2}+\frac{10x}{x^{2}+7}\)

Answer:

\(\frac{6}{x}+\frac{5}{x + 2}+\frac{10x}{x^{2}+7}\)