Calculus Examples

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

Step 1

Differentiate.

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

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

Step 2

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

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply $2$ by $- 3$ .

$3 x^{2} - 6 y x + \frac{d}{d x} [3 x y^{2}] + \frac{d}{d x} [- y^{3}]$

Step 3

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

Tap for more steps...

Step 3.1

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

$3 x^{2} - 6 y x + 3 y^{2} \frac{d}{d x} [x] + \frac{d}{d x} [- y^{3}]$

Step 3.2

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

$3 x^{2} - 6 y x + 3 y^{2} \cdot 1 + \frac{d}{d x} [- y^{3}]$

Step 3.3

Multiply $3$ by $1$ .

$3 x^{2} - 6 y x + 3 y^{2} + \frac{d}{d x} [- y^{3}]$

Step 4

Since $- y^{3}$ is constant with respect to $x$ , the derivative of $- y^{3}$ with respect to $x$ is $0$ .

$3 x^{2} - 6 y x + 3 y^{2} + 0$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Add $3 x^{2} - 6 y x + 3 y^{2}$ and $0$ .

$3 x^{2} - 6 y x + 3 y^{2}$

Step 5.2

Reorder terms.

$3 x^{2} + 3 y^{2} - 6 x y$