Question

differentiate the function.
y = (3x^2 - 9)^(-11)
\frac{dy}{dx}=-66x(3x^2 - 9)^(-12)

Explanation:

Step1: Identify the outer - function

Let $u = 3x^{2}-9$, then $y = u^{-11}$. The derivative of $y$ with respect to $u$ is $\frac{dy}{du}=-11u^{-12}$ by the power rule $\frac{d}{du}(u^{n})=nu^{n - 1}$.

Step2: Identify the inner - function

The derivative of $u = 3x^{2}-9$ with respect to $x$ is $\frac{du}{dx}=6x$ using the power rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$.

Step3: Apply the chain rule

The chain rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=-11u^{-12}$ and $\frac{du}{dx}=6x$ into the chain - rule formula. Replace $u$ with $3x^{2}-9$. So $\frac{dy}{dx}=-11(3x^{2}-9)^{-12}\cdot6x=-66x(3x^{2}-9)^{-12}$.

Answer:

$\frac{dy}{dx}=-66x(3x^{2}-9)^{-12}$