Question

replace ? with an expression that will make the equation valid.
$\frac{d}{dx}(6 - 5x^{2})^{7}=7(6 - 5x^{2})^{6}$ ?
the missing expression is

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, $u = 6-5x^{2}$ and $n = 7$. We know that $\frac{d}{dx}(u^{n})=n\cdot u^{n - 1}\cdot\frac{du}{dx}$, and we are given $\frac{d}{dx}(6 - 5x^{2})^{7}=7(6 - 5x^{2})^{6}\cdot?$.

Step2: Find $\frac{du}{dx}$

If $u = 6-5x^{2}$, then by the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, $\frac{du}{dx}=\frac{d}{dx}(6)-\frac{d}{dx}(5x^{2})$. Since $\frac{d}{dx}(6) = 0$ and $\frac{d}{dx}(5x^{2})=10x$, we have $\frac{du}{dx}=- 10x$.

Answer:

$-10x$