find the derivative of y with respect to x.\ny = (6x + 7)^x\n\frac{dy}{dx}=square\n(use parentheses to…

find the derivative of y with respect to x.\ny = (6x + 7)^x\n\frac{dy}{dx}=square\n(use parentheses to clearly denote the argument of each function.)

find the derivative of y with respect to x.\ny = (6x + 7)^x\n\frac{dy}{dx}=square\n(use parentheses to clearly denote the argument of each function.)

Answer

Explanation:

Step1: Take natural - log of both sides

$\ln y=x\ln(6x + 7)$

Step2: Differentiate both sides with respect to x

Using the product rule $(uv)^\prime = u^\prime v+uv^\prime$ where $u = x$ and $v=\ln(6x + 7)$. The derivative of $\ln y$ with respect to $x$ is $\frac{1}{y}\frac{dy}{dx}$, the derivative of $x$ is $1$, and the derivative of $\ln(6x + 7)$ using the chain - rule is $\frac{6}{6x + 7}$. So, $\frac{1}{y}\frac{dy}{dx}=1\times\ln(6x + 7)+x\times\frac{6}{6x + 7}$.

Step3: Solve for $\frac{dy}{dx}$

Multiply both sides by $y=(6x + 7)^x$. Then $\frac{dy}{dx}=(6x + 7)^x\left(\ln(6x + 7)+\frac{6x}{6x + 7}\right)$

Answer:

$(6x + 7)^x\left(\ln(6x + 7)+\frac{6x}{6x + 7}\right)$