Question

replace ? with an expression that will make the equation valid.
$\frac{d}{dx}(2x + 7)^5 = 5(2x + 7)^4$ ?

Explanation:

Step1: Apply chain - rule

The chain - rule states that if $y = u^n$ where $u$ is a function of $x$, then $\frac{dy}{dx}=n\cdot u^{n - 1}\cdot\frac{du}{dx}$. Here $u = 2x+7$ and $n = 5$.
For $y=(2x + 7)^5$, by the chain - rule $\frac{d}{dx}(2x + 7)^5=5(2x + 7)^4\cdot\frac{d}{dx}(2x + 7)$.

Step2: Differentiate inner function

We know that $\frac{d}{dx}(2x + 7)=\frac{d}{dx}(2x)+\frac{d}{dx}(7)$. Since $\frac{d}{dx}(ax)=a$ and $\frac{d}{dx}(c)=0$ for a constant $c$, then $\frac{d}{dx}(2x)=2$ and $\frac{d}{dx}(7)=0$. So $\frac{d}{dx}(2x + 7)=2$.

Answer:

$2$