Question

section 3.2 derivatives, exponential and logarithmic
question 10, 3.2.55
hw score: 47.37%, 9 of 19 points
points: 0 of 1
find $\frac{dy}{dx}$.
y = 3 ln x+8 log₃x
$\frac{dy}{dx}=square$ (type an exact answer.)

Explanation:

Step1: Recall derivative rules

The derivative of $\ln x$ is $\frac{1}{x}$, and for $\log_a x=\frac{\ln x}{\ln a}$, its derivative is $\frac{1}{x\ln a}$.

Step2: Differentiate term - by - term

For $y = 3\ln x+8\log_3 x$, the derivative of $3\ln x$ is $3\times\frac{1}{x}=\frac{3}{x}$ using the constant - multiple rule. The derivative of $8\log_3 x$ is $8\times\frac{1}{x\ln 3}=\frac{8}{x\ln 3}$.

Step3: Combine the derivatives

$\frac{dy}{dx}=\frac{3}{x}+\frac{8}{x\ln 3}$.

Answer:

$\frac{3}{x}+\frac{8}{x\ln 3}$