Question

What is the derivative of $x{y}^3$?

Answer 1

Step 1: Calculating differentiate the function $x{y}^3$ with respect to $x$

Derivative formula:

Power rule:$\frac{d{x}^a}{dx}=a{x}^{a-1}$

Composite function: $\frac{d}{dx}\left(f\left(g\left(x\right)\right)\right)=f\text{'}\left(g\left(x\right)\right)·g\text{'}\left(x\right)$

Product rule: $\frac{d}{dx}\left(uv\right)=u\frac{dv}{dx}+v\frac{du}{dx}$

Differentiate the function $x{y}^3$ with respect to $x$, we get

$\frac{d}{dx}x{y}^3=x\left(\frac{d{y}^3}{dx}\right)+y^3\left(\frac{dx}{dx}\right)$

$\Rightarrow$$\frac{d}{dx}x{y}^3=x\left(3y^2\right)\frac{dy}{dx}+y^3\left(1\right)$

$\Rightarrow$$\frac{d}{dx}x{y}^3=3x{y}^2\frac{dy}{dx}+y^3$

Step 2: Calculating differentiate the function $x{y}^3$ with respect to $y$

Differentiate the function $x{y}^3$ with respect to $y$, we get

$\frac{d}{dy}x{y}^3=x\left(\frac{d{y}^3}{dy}\right)+y^3\left(\frac{dx}{dy}\right)$

$\Rightarrow \frac{d}{dy}x{y}^3=x\left(3y^2\right)+y^3\frac{dx}{dy}$

$\Rightarrow \frac{d}{dy}x{y}^3=3x{y}^2+y^3\frac{dx}{dy}$

Hence, the derivative of $x{y}^3$ with respect to $x$ is $3x{y}^2\frac{dy}{dx}+y^3$ and the derivative of $x{y}^3$ with respect to $y$ is $3x{y}^2+y^3\frac{dx}{dy}$.