question\ngiven the function $y = 4(-4x^{2}+7x - 9)^{6}$, find $\frac{dy}{dx}$ in any form.

question\ngiven the function $y = 4(-4x^{2}+7x - 9)^{6}$, find $\frac{dy}{dx}$ in any form.

question\ngiven the function $y = 4(-4x^{2}+7x - 9)^{6}$, find $\frac{dy}{dx}$ in any form.

Answer

Explanation:

Step1: Identify the outer - inner functions

Let $u=-4x^{2}+7x - 9$, then $y = 4u^{6}$.

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

Using the power rule $\frac{d}{du}(au^{n})=nau^{n - 1}$, for $y = 4u^{6}$, we have $\frac{dy}{du}=4\times6u^{5}=24u^{5}$.

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

For $u=-4x^{2}+7x - 9$, $\frac{du}{dx}=-8x + 7$.

Step4: Apply the chain - rule

The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=24u^{5}$ and $\frac{du}{dx}=-8x + 7$ into the chain - rule formula, and replace $u=-4x^{2}+7x - 9$. So $\frac{dy}{dx}=24(-4x^{2}+7x - 9)^{5}(-8x + 7)$.

Answer:

$24(-8x + 7)(-4x^{2}+7x - 9)^{5}$