Find the derivative: frac{d}{dx}[(log_e x)(lo | vedclass

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(D) We need to find the derivative of the product of two logarithmic functions: $\frac{d}{dx}[(\log_e x)(\log_a x)]$. First,use the change of base for

Vedclass · Accepted Answer

$\frac{2\log_a x}{x}$

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(D) We need to find the derivative of the product of two logarithmic functions: $\frac{d}{dx}[(\log_e x)(\log_a x)]$. First,use the change of base formula for logarithms: $\log_a x = \frac{\log_e x}{\log_e a}$. Substituting this into the expression: $\frac{d}{dx}[(\log_e x) \cdot \frac{\log_e x}{\log_e a}] = \frac{1}{\log_e a} \cdot \frac{d}{dx}[(\log_e x)^2]$. Using the chain rule,the derivative of $(\log_e x)^2$ is $2(\log_e x) \cdot \frac{d}{dx}(\log_e x) = 2(\log_e x) \cdot \frac{1}{x}$. Thus,the expression becomes: $\frac{1}{\log_e a} \cdot \frac{2\log_e x}{x} = \frac{2}{\log_e a} \cdot \frac{\log_e x}{x}$. Since $\frac{\log_e x}{\log_e a} = \log_a x$,the final result is $\frac{2\log_a x}{x}$.

Find the derivative: $\frac{d}{dx}[(\log_e x)(\log_a x)]$

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