In Calculus: if we apply chain rule to $\frac{d}{dx} (y^4)$, why do we only apply the product rule on $\frac{d}{dx} (x^2y)$?

Question

If we apply chain rule to $\frac{d}{dx} (y^4)$, because there's a function within a function.

Why do we only apply the product rule on $\frac{d}{dx} (x^2y)$?

Answer 1

The first function can be with the following functions: $$ f(x) = x^4$$

$$ g(y) = y$$

As:

$$ f( g) = y^4$$

Hence,

$$ \frac{df(g)}{dx} = \frac{df}{dx}|_{g(y)} \cdot \frac{dg(y)}{dx}$$

Then,

$$ \frac{df(g)}{dx} = 4y^3 \cdot \frac{dy}{dx}$$

The other example:

$$ \frac{d}{dx}(x^2 y) = y \frac{d}{dx} x^2 + x^2 \frac{d}{dx} y $$

Since

$$ \frac{d}{dx} y = \frac{dy}{dx}$$

There is no reason to chain rule.

Answer 2

The product rule $$ \frac {\mathrm d (u(x)v(x))}{\mathrm dx}=u'(x)v(x)+u(x)v'(x)$$ is the chain rule $$D(f\circ g)=Df\cdot Dg$$ applied to $g(x)=(u(x),v(x))$ and $f(x_1,x_2)=x_1x_2$.