Question

If $y$is a function of $x$, then how is the chain rule involved in differentiating $y^3$ with respect to $x$, and why?

Short answer

Derivative of $y^3$is $3y^2\times \frac{dy}{dx}$.

The chain rule is involved in differentiation because$y^3$is a composite function.

Step-by-step solution

Step 1. Given information

Function is$f\left(x\right)={\left(y\left(x\right)\right)}^3$

Step 2. To find derivative of y3 using chain rule.

$$\begin{gathered} f\left(x\right)={\left(y\right)}^3 \\ f\text{'}\left(x\right)=\frac{d}{dy}\left(y^3\right)\times \frac{dy}{dx} \\ f\text{'}\left(x\right)=3y^2\times \frac{dy}{dx} \end{gathered}$$

Since$y^3$is a function of $y$and $y$is a function of $x$,

so the $y^3$is a composite function.

Hence, the chain rule is applicable here.