Calculus Examples

$f (x) = 3 {(x + 1)}^{2}$

Step 1

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

$\frac{d}{d x} [3 ((x + 1) (x + 1))]$

Step 2

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

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Step 2.1

Apply the distributive property.

$\frac{d}{d x} [3 (x (x + 1) + 1 (x + 1))]$

Step 2.2

Apply the distributive property.

$\frac{d}{d x} [3 (x \cdot x + x \cdot 1 + 1 (x + 1))]$

Step 2.3

Apply the distributive property.

$\frac{d}{d x} [3 (x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1)]$

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 $x$ .

$\frac{d}{d x} [3 (x^{2} + x \cdot 1 + 1 x + 1 \cdot 1)]$

Step 3.1.2

Multiply $x$ by $1$ .

$\frac{d}{d x} [3 (x^{2} + x + 1 x + 1 \cdot 1)]$

Step 3.1.3

Multiply $x$ by $1$ .

$\frac{d}{d x} [3 (x^{2} + x + x + 1 \cdot 1)]$

Step 3.1.4

Multiply $1$ by $1$ .

$\frac{d}{d x} [3 (x^{2} + x + x + 1)]$

Step 3.2

Add $x$ and $x$ .

$\frac{d}{d x} [3 (x^{2} + 2 x + 1)]$

Step 4

Since $3$ is constant with respect to $x$ , the derivative of $3 (x^{2} + 2 x + 1)$ with respect to $x$ is $3 \frac{d}{d x} [x^{2} + 2 x + 1]$ .

$3 \frac{d}{d x} [x^{2} + 2 x + 1]$

Step 5

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

$3 (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [1])$

Step 6

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

$3 (2 x + \frac{d}{d x} [2 x] + \frac{d}{d x} [1])$

Step 7

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

$3 (2 x + 2 \frac{d}{d x} [x] + \frac{d}{d x} [1])$

Step 8

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

$3 (2 x + 2 \cdot 1 + \frac{d}{d x} [1])$

Step 9

Multiply $2$ by $1$ .

$3 (2 x + 2 + \frac{d}{d x} [1])$

Step 10

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

$3 (2 x + 2 + 0)$

Step 11

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

$3 (2 x + 2)$

Step 12

Simplify.

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Step 12.1

Apply the distributive property.

$3 (2 x) + 3 \cdot 2$

Step 12.2

Combine terms.

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Step 12.2.1

Multiply $2$ by $3$ .

$6 x + 3 \cdot 2$

Step 12.2.2

Multiply $3$ by $2$ .

$6 x + 6$