Calculus Examples

$z = x y^{3} + x^{3} y^{2}$

Step 1

This derivative could not be completed using the chain rule. Mathway will use another method.

Step 2

By the Sum Rule, the derivative of $x y^{3} + x^{3} y^{2}$ with respect to $y$ is $\frac{d}{d y} [x y^{3}] + \frac{d}{d y} [x^{3} y^{2}]$ .

$\frac{d}{d y} [x y^{3}] + \frac{d}{d y} [x^{3} y^{2}]$

Step 3

Evaluate $\frac{d}{d y} [x y^{3}]$ .

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Step 3.1

Since $x$ is constant with respect to $y$ , the derivative of $x y^{3}$ with respect to $y$ is $x \frac{d}{d y} [y^{3}]$ .

$x \frac{d}{d y} [y^{3}] + \frac{d}{d y} [x^{3} y^{2}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 3$ .

$x (3 y^{2}) + \frac{d}{d y} [x^{3} y^{2}]$

Step 3.3

Move $3$ to the left of $x$ .

$3 x y^{2} + \frac{d}{d y} [x^{3} y^{2}]$

Step 4

Evaluate $\frac{d}{d y} [x^{3} y^{2}]$ .

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Step 4.1

Since $x^{3}$ is constant with respect to $y$ , the derivative of $x^{3} y^{2}$ with respect to $y$ is $x^{3} \frac{d}{d y} [y^{2}]$ .

$3 x y^{2} + x^{3} \frac{d}{d y} [y^{2}]$

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 2$ .

$3 x y^{2} + x^{3} (2 y)$

Step 4.3

Move $2$ to the left of $x^{3}$ .

$3 x y^{2} + 2 x^{3} y$

Step 5

Reorder terms.

$3 y^{2} x + 2 x^{3} y$