Question

find $\frac{dy}{dx}$
$y = 7ln x+8log_7x$
$\frac{dy}{dx}=square$ (type an exact answer.)

Explanation:

Step1: Recall derivative of $\ln x$

The derivative of $\ln x$ with respect to $x$ is $\frac{1}{x}$. So the derivative of $7\ln x$ is $7\times\frac{1}{x}=\frac{7}{x}$.

Step2: Recall change - of - base formula and derivative of $\log_a x$

The change - of - base formula for $\log_a x=\frac{\ln x}{\ln a}$. So $\log_7 x = \frac{\ln x}{\ln 7}$. Then the derivative of $8\log_7 x$ is $8\times\frac{1}{x\ln 7}=\frac{8}{x\ln 7}$.

Step3: Use sum rule of derivatives

If $y = u + v$, then $\frac{dy}{dx}=\frac{du}{dx}+\frac{dv}{dx}$. Here $u = 7\ln x$ and $v = 8\log_7 x$. So $\frac{dy}{dx}=\frac{7}{x}+\frac{8}{x\ln 7}$.

Answer:

$\frac{7}{x}+\frac{8}{x\ln 7}$