Question

differentiate the function.
y = (4x^2 - 14)^(-11)
\frac{dy}{dx}=square

Explanation:

Step1: Identify the outer - inner functions

Let $u = 4x^{2}-14$, so $y = u^{-11}$.

Step2: Differentiate the outer function

The derivative of $y$ with respect to $u$ is $\frac{dy}{du}=-11u^{-12}$ using the power rule $\frac{d}{du}(u^{n})=nu^{n - 1}$.

Step3: Differentiate the inner function

The derivative of $u$ with respect to $x$ is $\frac{du}{dx}=8x$ since $\frac{d}{dx}(4x^{2}-14)=4\times2x-0 = 8x$.

Step4: Apply the chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}$ and $\frac{du}{dx}$:
$\frac{dy}{dx}=-11u^{-12}\cdot8x$.
Replace $u$ with $4x^{2}-14$:
$\frac{dy}{dx}=-88x(4x^{2}-14)^{-12}$.

Answer:

$-88x(4x^{2}-14)^{-12}$