Question

find $\frac{dy}{dx}$
$y = 5^{log 2x}$
$\frac{dy}{dx}=square$ (simplify your answer.)

Explanation:

Step1: Recall derivative formula for $a^u$

The derivative of $a^u$ with respect to $x$ is $a^u\ln a\cdot\frac{du}{dx}$, where $a = 5$ and $u=\log_2x$.

Step2: Find $\frac{du}{dx}$

The derivative of $\log_2x$ with respect to $x$ is $\frac{1}{x\ln 2}$ using the formula $\frac{d}{dx}\log_ax=\frac{1}{x\ln a}$.

Step3: Calculate $\frac{dy}{dx}$

Substitute $a = 5$, $u=\log_2x$ and $\frac{du}{dx}=\frac{1}{x\ln 2}$ into the formula for $\frac{d}{dx}(a^u)$. We get $\frac{dy}{dx}=5^{\log_2x}\ln 5\cdot\frac{1}{x\ln 2}$.

Answer:

$\frac{5^{\log_2x}\ln 5}{x\ln 2}$