Question

find the derivative of $f(x) = \ln\left(\frac{x^6}{2x^5 - 7}\
ight)$.\
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$\boldsymbol{f(x) = \frac{6}{x} + \frac{10x^4}{2x^5 - 7}}$\
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$\boldsymbol{f(x) = \frac{6}{x} - \frac{1}{2x^5 - 7}}$\
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$\boldsymbol{f(x) = 6\ln(x) - \ln(2x^5 - 7)}$\
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$\boldsymbol{f(x) = \frac{6}{x} - \frac{10x^4}{2x^5 - 7}}$

Explanation:

Step1: Simplify using log quotient rule

$f(x) = \ln(x^6) - \ln(2x^5 - 7)$

Step2: Simplify $\ln(x^6)$ with power rule

$f(x) = 6\ln(x) - \ln(2x^5 - 7)$

Step3: Differentiate term by term

Derivative of $6\ln(x)$ is $\frac{6}{x}$. For $\ln(2x^5 -7)$, use chain rule: $\frac{1}{2x^5 -7} \cdot 10x^4$.

Step4: Combine derivatives

$f'(x) = \frac{6}{x} - \frac{10x^4}{2x^5 -7}$

Answer:

$f'(x)=\frac{6}{x}-\frac{10x^4}{2x^5 - 7}$