replace ? with an expression that will make the equation valid. $\frac{d}{dx}(7 - 4x^{2})^{9}=9(7…

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 $u = 7-4x^{2}$ and $n = 9$. So $\frac{d}{dx}(7 - 4x^{2})^{9}=9(7 - 4x^{2})^{8}\cdot\frac{d}{dx}(7 - 4x^{2})$.

Step2: Differentiate $7 - 4x^{2}$

We know that $\frac{d}{dx}(7-4x^{2})=\frac{d}{dx}(7)-4\frac{d}{dx}(x^{2})$. Since $\frac{d}{dx}(c)=0$ for a constant $c$ and $\frac{d}{dx}(x^{n})=nx^{n - 1}$, then $\frac{d}{dx}(7)=0$ and $\frac{d}{dx}(x^{2}) = 2x$. So $\frac{d}{dx}(7-4x^{2})=-8x$.

Answer:

$-8x$