Calculus Examples

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

Step 1

Rewrite ${(x + y)}^{2}$ as $(x + y) (x + y)$ .

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

Step 2

Expand $(x + y) (x + y)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

$\frac{d}{d x} [x \cdot x + x y + y (x + y)]$

Step 2.3

Apply the distributive property.

$\frac{d}{d x} [x \cdot x + x y + y x + y \cdot y]$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

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

Step 3.1.2

Multiply $y$ by $y$ .

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

Step 3.2

Add $x y$ and $y x$ .

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Step 3.2.1

Reorder $y$ and $x$ .

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

Step 3.2.2

Add $x y$ and $x y$ .

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Multiply $2$ by $1$ .

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

Step 9

Since $y^{2}$ is constant with respect to $x$ , the derivative of $y^{2}$ with respect to $x$ is $0$ .

$2 x + 2 y + 0$

Step 10

Add $2 x + 2 y$ and $0$ .

$2 x + 2 y$