Question

find the derivative of the function.
y = - 9(9x^{2}+5)^{-6}

\frac{dy}{dx}=\square
(type an expression using x as the variable.)

Explanation:

Step1: Identify the outer - inner functions

Let $u = 9x^{2}+5$, then $y=-9u^{-6}$.

Step2: Differentiate the outer function with respect to $u$

Using the power rule $\frac{d}{du}(au^{n})=nau^{n - 1}$, we have $\frac{dy}{du}=-9\times(-6)u^{-7}=54u^{-7}$.

Step3: Differentiate the inner function with respect to $x$

$\frac{du}{dx}=\frac{d}{dx}(9x^{2}+5)=18x$.

Step4: Apply the chain - rule

The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}$ and $\frac{du}{dx}$ into the chain - rule formula: $\frac{dy}{dx}=54u^{-7}\cdot18x$.

Step5: Substitute $u = 9x^{2}+5$ back in

$\frac{dy}{dx}=54(9x^{2}+5)^{-7}\cdot18x$.
Simplify the expression: $\frac{dy}{dx}=972x(9x^{2}+5)^{-7}=\frac{972x}{(9x^{2}+5)^{7}}$.

Answer:

$\frac{972x}{(9x^{2}+5)^{7}}$