calculate the derivative of the following function\n$y = 5(7x^{2}+6)^{-4}$\n$\frac{dy}{dx}=square$

Explanation:

Step1: Identify outer - inner functions

Let $u = 7x^{2}+6$, then $y = 5u^{-4}$.

Step2: Differentiate outer function

Using the power rule $\frac{d}{du}(au^{n})=anu^{n - 1}$, for $y = 5u^{-4}$, $\frac{dy}{du}=5\times(-4)u^{-5}=-20u^{-5}$.

Step3: Differentiate inner function

For $u = 7x^{2}+6$, $\frac{du}{dx}=14x$.

Step4: Apply chain - rule

The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}$ and $\frac{du}{dx}$: $\frac{dy}{dx}=-20u^{-5}\cdot14x$.

Step5: Substitute back $u$

Replace $u = 7x^{2}+6$ into the above expression: $\frac{dy}{dx}=-20(7x^{2}+6)^{-5}\cdot14x=-280x(7x^{2}+6)^{-5}$.

Answer:

$-280x(7x^{2}+6)^{-5}$