Question

calculate the derivative of the following function. y = \sqrt{2x + 13} \frac{dy}{dx}=\square

Explanation:

Step1: Rewrite the function

Rewrite $y = \sqrt{2x + 13}$ as $y=(2x + 13)^{\frac{1}{2}}$.

Step2: Apply chain - rule

The chain - rule states that if $y = u^n$ and $u$ is a function of $x$, then $\frac{dy}{dx}=n\cdot u^{n - 1}\cdot\frac{du}{dx}$. Here, $n=\frac{1}{2}$ and $u = 2x+13$, and $\frac{du}{dx}=2$.
So, $\frac{dy}{dx}=\frac{1}{2}(2x + 13)^{\frac{1}{2}-1}\cdot2$.

Step3: Simplify the expression

First, $\frac{1}{2}-1=-\frac{1}{2}$. Then $\frac{dy}{dx}=\frac{1}{2}(2x + 13)^{-\frac{1}{2}}\cdot2$.
The $\frac{1}{2}$ and $2$ cancel out, and we get $\frac{dy}{dx}=(2x + 13)^{-\frac{1}{2}}=\frac{1}{\sqrt{2x + 13}}$.

Answer:

$\frac{1}{\sqrt{2x + 13}}$