Question

find the derivative of y with respect to x.
y = ln(11x) + 5x
dy/dx = □

Explanation:

Step1: Apply sum - rule of derivatives

The derivative of a sum $u + v$ is $\frac{d(u + v)}{dx}=\frac{du}{dx}+\frac{dv}{dx}$. Here $u = \ln(11x)$ and $v = 5x$. So $\frac{dy}{dx}=\frac{d}{dx}(\ln(11x))+\frac{d}{dx}(5x)$.

Step2: Differentiate $\ln(11x)$

Using the chain - rule, if $y=\ln(u)$ and $u = 11x$, then $\frac{dy}{du}=\frac{1}{u}$ and $\frac{du}{dx}=11$. By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. So $\frac{d}{dx}(\ln(11x))=\frac{1}{11x}\cdot11=\frac{1}{x}$.

Step3: Differentiate $5x$

The power - rule states that if $y = ax^n$, then $\frac{dy}{dx}=anx^{n - 1}$. For $y = 5x$ ($a = 5$, $n = 1$), $\frac{d}{dx}(5x)=5$.

Step4: Combine the results

$\frac{dy}{dx}=\frac{1}{x}+5$.

Answer:

$\frac{1}{x}+5$