Calculus Examples

$y = {(2 x + 1)}^{2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ({(2 x + 1)}^{2})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Rewrite ${(2 x + 1)}^{2}$ as $(2 x + 1) (2 x + 1)$ .

$\frac{d}{d x} [(2 x + 1) (2 x + 1)]$

Step 3.2

Expand $(2 x + 1) (2 x + 1)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$\frac{d}{d x} [2 x (2 x + 1) + 1 (2 x + 1)]$

Step 3.2.2

Apply the distributive property.

$\frac{d}{d x} [2 x (2 x) + 2 x \cdot 1 + 1 (2 x + 1)]$

Step 3.2.3

Apply the distributive property.

$\frac{d}{d x} [2 x (2 x) + 2 x \cdot 1 + 1 (2 x) + 1 \cdot 1]$

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [2 \cdot 2 x \cdot x + 2 x \cdot 1 + 1 (2 x) + 1 \cdot 1]$

Step 3.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.3.1.2.1

Move $x$ .

$\frac{d}{d x} [2 \cdot 2 (x \cdot x) + 2 x \cdot 1 + 1 (2 x) + 1 \cdot 1]$

Step 3.3.1.2.2

Multiply $x$ by $x$ .

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

Step 3.3.1.3

Multiply $2$ by $2$ .

$\frac{d}{d x} [4 x^{2} + 2 x \cdot 1 + 1 (2 x) + 1 \cdot 1]$

Step 3.3.1.4

Multiply $2$ by $1$ .

$\frac{d}{d x} [4 x^{2} + 2 x + 1 (2 x) + 1 \cdot 1]$

Step 3.3.1.5

Multiply $2 x$ by $1$ .

$\frac{d}{d x} [4 x^{2} + 2 x + 2 x + 1 \cdot 1]$

Step 3.3.1.6

Multiply $1$ by $1$ .

$\frac{d}{d x} [4 x^{2} + 2 x + 2 x + 1]$

Step 3.3.2

Add $2 x$ and $2 x$ .

$\frac{d}{d x} [4 x^{2} + 4 x + 1]$

Step 3.4

By the Sum Rule, the derivative of $4 x^{2} + 4 x + 1$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [1]$

Step 3.5

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [1]$

Step 3.6

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

$4 (2 x) + \frac{d}{d x} [4 x] + \frac{d}{d x} [1]$

Step 3.7

Multiply $2$ by $4$ .

$8 x + \frac{d}{d x} [4 x] + \frac{d}{d x} [1]$

Step 3.8

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$8 x + 4 \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 3.9

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

$8 x + 4 \cdot 1 + \frac{d}{d x} [1]$

Step 3.10

Multiply $4$ by $1$ .

$8 x + 4 + \frac{d}{d x} [1]$

Step 3.11

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$8 x + 4 + 0$

Step 3.12

Add $8 x + 4$ and $0$ .

$8 x + 4$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 8 x + 4$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 8 x + 4$