Answer
$\frac{dy}{dx}=-12x^3(5-x^4)^2$
Work Step by Step
Considering $u=5-x^4$, the function can be written as $y=u^3$.
We have the composite function $y=f(g(x))$ where $g(x)=5-x^4$ and $f(u)=u^3$.
Find $\frac{dy}{du}$:
$\frac{dy}{du}=\frac{d}{du}(u^3)=3u^2$
Find $\frac{du}{dx}$:
$\frac{du}{dx}=\frac{d}{dx}(5-x^4)=0-4x^3=-4x^3$
Find $\frac{dy}{dx}$ using the chain rule:
$\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}=3u^2\cdot (-4x^3)=3(5-x^4)^2\cdot (-4x^3)=-12x^3(5-x^4)^2$
Thus,
$\frac{dy}{dx}=-12x^3(5-x^4)^2$
