Question

question
find $\frac{dy}{dx}$ if $y=(2x^{4}+1)^{-5}$.
provide your answer below:
$\frac{dy}{dx}=square$

Explanation:

Step1: Apply chain - rule

The chain - rule states that if $y = u^{-5}$ and $u = 2x^{4}+1$, then $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. First, find $\frac{dy}{du}$. Using the power rule $\frac{d}{du}(u^{n})=nu^{n - 1}$, for $y = u^{-5}$, we have $\frac{dy}{du}=-5u^{-6}$.

Step2: Find $\frac{du}{dx}$

For $u = 2x^{4}+1$, using the power rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we get $\frac{du}{dx}=8x^{3}$.

Step3: Calculate $\frac{dy}{dx}$

Substitute $u = 2x^{4}+1$ back into $\frac{dy}{du}$ and then multiply by $\frac{du}{dx}$. So, $\frac{dy}{dx}=-5(2x^{4}+1)^{-6}\cdot8x^{3}=-40x^{3}(2x^{4}+1)^{-6}$.

Answer:

$-40x^{3}(2x^{4}+1)^{-6}$