use logarithmic differentiation to evaluate y. y = (10x)^(ln 10x) y = □ (use parentheses to clearly denote…

Explanation:

Step1: Take natural - log of both sides

$\ln y=\ln((10x)^{\ln 10x})=\ln 10x\cdot\ln(10x)=(\ln 10x)^2$

Step2: Differentiate both sides with respect to $x$

Using the chain - rule, the derivative of the left - hand side is $\frac{y'}{y}$. For the right - hand side, let $u = \ln 10x$, then the function is $u^{2}$. The derivative of $u^{2}$ with respect to $u$ is $2u$, and the derivative of $u=\ln 10x$ with respect to $x$ is $\frac{1}{x}$. So the derivative of $(\ln 10x)^2$ with respect to $x$ is $2\ln 10x\cdot\frac{1}{x}$. We have $\frac{y'}{y}=\frac{2\ln 10x}{x}$

Step3: Solve for $y'$

Multiply both sides by $y$. Since $y=(10x)^{\ln 10x}$, then $y'=(10x)^{\ln 10x}\cdot\frac{2\ln 10x}{x}$

Answer:

$(10x)^{\ln 10x}\cdot\frac{2\ln 10x}{x}$