Calculus Examples

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

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x + 3$ and $g (x) = x^{2} - 3$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

$(x + 3) (2 x + 0) + (x^{2} - 3) \frac{d}{d x} [x + 3]$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $2 x$ and $0$ .

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

Step 2.4.2

Move $2$ to the left of $x + 3$ .

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$2 (x + 3) x + (x^{2} - 3) (1 + 0)$

Step 2.8

Simplify the expression.

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Step 2.8.1

Add $1$ and $0$ .

$2 (x + 3) x + (x^{2} - 3) \cdot 1$

Step 2.8.2

Multiply $x^{2} - 3$ by $1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$(2 x + 2 \cdot 3) x + x^{2} - 3$

Step 3.2

Apply the distributive property.

$2 x \cdot x + 2 \cdot 3 x + x^{2} - 3$

Step 3.3

Combine terms.

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Step 3.3.1

Raise $x$ to the power of $1$ .

$2 (x^{1} x) + 2 \cdot 3 x + x^{2} - 3$

Step 3.3.2

Raise $x$ to the power of $1$ .

$2 (x^{1} x^{1}) + 2 \cdot 3 x + x^{2} - 3$

Step 3.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x^{1 + 1} + 2 \cdot 3 x + x^{2} - 3$

Step 3.3.4

Add $1$ and $1$ .

$2 x^{2} + 2 \cdot 3 x + x^{2} - 3$

Step 3.3.5

Multiply $2$ by $3$ .

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

Step 3.3.6

Add $2 x^{2}$ and $x^{2}$ .

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