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webwork / math220f25 / derivatives of logarithmic functions / 2
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derivatives of logarithmic functions
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let ( f(x)=x^{2x}).
use logarithmic differentiation to determine the derivative.
( f(x)=)
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Explanation:

Step1: Take natural - log on both sides

Let $y = x^{2x}$. Then $\ln y=\ln(x^{2x}) = 2x\ln x$.

Step2: Differentiate both sides with respect to $x$

Using the product rule $(uv)^\prime = u^\prime v+uv^\prime$ where $u = 2x$ and $v=\ln x$. The derivative of $\ln y$ with respect to $x$ is $\frac{y^\prime}{y}$, the derivative of $2x$ is $2$ and the derivative of $\ln x$ is $\frac{1}{x}$. So $\frac{y^\prime}{y}=2\ln x + 2x\cdot\frac{1}{x}=2\ln x + 2$.

Step3: Solve for $y^\prime$

Multiply both sides by $y$. Since $y = x^{2x}$, we have $y^\prime=x^{2x}(2\ln x + 2)$.

Answer:

$x^{2x}(2\ln x + 2)$