Calculus Examples

$[(x^{2} + x^{3}) (x + 9)]$

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 $f (x) = x^{2} + x^{3}$ and $g (x) = x + 9$ .

$(x^{2} + x^{3}) \frac{d}{d x} [x + 9] + (x + 9) \frac{d}{d x} [x^{2} + x^{3}]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x + 9$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [9]$ .

$(x^{2} + x^{3}) (\frac{d}{d x} [x] + \frac{d}{d x} [9]) + (x + 9) \frac{d}{d x} [x^{2} + x^{3}]$

Step 2.2

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

$(x^{2} + x^{3}) (1 + \frac{d}{d x} [9]) + (x + 9) \frac{d}{d x} [x^{2} + x^{3}]$

Step 2.3

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$(x^{2} + x^{3}) (1 + 0) + (x + 9) \frac{d}{d x} [x^{2} + x^{3}]$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $1$ and $0$ .

$(x^{2} + x^{3}) \cdot 1 + (x + 9) \frac{d}{d x} [x^{2} + x^{3}]$

Step 2.4.2

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

$x^{2} + x^{3} + (x + 9) \frac{d}{d x} [x^{2} + x^{3}]$

Step 2.5

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

$x^{2} + x^{3} + (x + 9) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [x^{3}])$

Step 2.6

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

$x^{2} + x^{3} + (x + 9) (2 x + \frac{d}{d x} [x^{3}])$

Step 2.7

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

$x^{2} + x^{3} + (x + 9) (2 x + 3 x^{2})$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$x^{2} + x^{3} + (x + 9) (2 x) + (x + 9) (3 x^{2})$

Step 3.2

Apply the distributive property.

$x^{2} + x^{3} + x (2 x) + 9 (2 x) + (x + 9) (3 x^{2})$

Step 3.3

Apply the distributive property.

$x^{2} + x^{3} + x (2 x) + 9 (2 x) + x (3 x^{2}) + 9 (3 x^{2})$

Step 3.4

Combine terms.

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Step 3.4.1

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

$x^{2} + x^{3} + 2 (x^{1} x) + 9 (2 x) + x (3 x^{2}) + 9 (3 x^{2})$

Step 3.4.2

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

$x^{2} + x^{3} + 2 (x^{1} x^{1}) + 9 (2 x) + x (3 x^{2}) + 9 (3 x^{2})$

Step 3.4.3

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

$x^{2} + x^{3} + 2 x^{1 + 1} + 9 (2 x) + x (3 x^{2}) + 9 (3 x^{2})$

Step 3.4.4

Add $1$ and $1$ .

$x^{2} + x^{3} + 2 x^{2} + 9 (2 x) + x (3 x^{2}) + 9 (3 x^{2})$

Step 3.4.5

Multiply $2$ by $9$ .

$x^{2} + x^{3} + 2 x^{2} + 18 x + x (3 x^{2}) + 9 (3 x^{2})$

Step 3.4.6

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

$x^{2} + x^{3} + 2 x^{2} + 18 x + 3 (x^{1} x^{2}) + 9 (3 x^{2})$

Step 3.4.7

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

$x^{2} + x^{3} + 2 x^{2} + 18 x + 3 x^{1 + 2} + 9 (3 x^{2})$

Step 3.4.8

Add $1$ and $2$ .

$x^{2} + x^{3} + 2 x^{2} + 18 x + 3 x^{3} + 9 (3 x^{2})$

Step 3.4.9

Multiply $3$ by $9$ .

$x^{2} + x^{3} + 2 x^{2} + 18 x + 3 x^{3} + 27 x^{2}$

Step 3.4.10

Add $2 x^{2}$ and $27 x^{2}$ .

$x^{2} + x^{3} + 29 x^{2} + 18 x + 3 x^{3}$

Step 3.4.11

Add $x^{2}$ and $29 x^{2}$ .

$x^{3} + 30 x^{2} + 18 x + 3 x^{3}$

Step 3.4.12

Add $x^{3}$ and $3 x^{3}$ .

$4 x^{3} + 30 x^{2} + 18 x$