Question

let $f(x)=x^{5x}$. use logarithmic differentiation to determine the derivative. $f(x)=$

Explanation:

Step1: Take natural - log of both sides

$\ln(f(x))=\ln(x^{5x})$. Using the property $\ln(a^b)=b\ln(a)$, we get $\ln(f(x)) = 5x\ln(x)$.

Step2: Differentiate both sides with respect to $x$

The derivative of the left - hand side is $\frac{f^{\prime}(x)}{f(x)}$ by the chain rule. For the right - hand side, use the product rule $(uv)^\prime = u^\prime v+uv^\prime$, where $u = 5x$ and $v=\ln(x)$. $u^\prime=5$ and $v^\prime=\frac{1}{x}$. So, $(5x\ln(x))^\prime=5\ln(x)+5x\cdot\frac{1}{x}=5\ln(x) + 5$.

Step3: Solve for $f^{\prime}(x)$

Since $\frac{f^{\prime}(x)}{f(x)}=5\ln(x)+5$ and $f(x)=x^{5x}$, then $f^{\prime}(x)=x^{5x}(5\ln(x)+5)$.

Answer:

$x^{5x}(5\ln(x)+5)$