Calculus Examples

$\frac{11 x^{2} + 10 x - 9}{x + 9}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 11 x^{2} + 10 x - 9$ and $g (x) = x + 9$ .

$\frac{(x + 9) \frac{d}{d x} [11 x^{2} + 10 x - 9] - (11 x^{2} + 10 x - 9) \frac{d}{d x} [x + 9]}{{(x + 9)}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $11 x^{2} + 10 x - 9$ with respect to $x$ is $\frac{d}{d x} [11 x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [- 9]$ .

$\frac{(x + 9) (\frac{d}{d x} [11 x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [- 9]) - (11 x^{2} + 10 x - 9) \frac{d}{d x} [x + 9]}{{(x + 9)}^{2}}$

Step 2.2

Since $11$ is constant with respect to $x$ , the derivative of $11 x^{2}$ with respect to $x$ is $11 \frac{d}{d x} [x^{2}]$ .

$\frac{(x + 9) (11 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [- 9]) - (11 x^{2} + 10 x - 9) \frac{d}{d x} [x + 9]}{{(x + 9)}^{2}}$

Step 2.3

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

$\frac{(x + 9) (11 (2 x) + \frac{d}{d x} [10 x] + \frac{d}{d x} [- 9]) - (11 x^{2} + 10 x - 9) \frac{d}{d x} [x + 9]}{{(x + 9)}^{2}}$

Step 2.4

Multiply $2$ by $11$ .

$\frac{(x + 9) (22 x + \frac{d}{d x} [10 x] + \frac{d}{d x} [- 9]) - (11 x^{2} + 10 x - 9) \frac{d}{d x} [x + 9]}{{(x + 9)}^{2}}$

Step 2.5

Since $10$ is constant with respect to $x$ , the derivative of $10 x$ with respect to $x$ is $10 \frac{d}{d x} [x]$ .

$\frac{(x + 9) (22 x + 10 \frac{d}{d x} [x] + \frac{d}{d x} [- 9]) - (11 x^{2} + 10 x - 9) \frac{d}{d x} [x + 9]}{{(x + 9)}^{2}}$

Step 2.6

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

$\frac{(x + 9) (22 x + 10 \cdot 1 + \frac{d}{d x} [- 9]) - (11 x^{2} + 10 x - 9) \frac{d}{d x} [x + 9]}{{(x + 9)}^{2}}$

Step 2.7

Multiply $10$ by $1$ .

$\frac{(x + 9) (22 x + 10 + \frac{d}{d x} [- 9]) - (11 x^{2} + 10 x - 9) \frac{d}{d x} [x + 9]}{{(x + 9)}^{2}}$

Step 2.8

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

$\frac{(x + 9) (22 x + 10 + 0) - (11 x^{2} + 10 x - 9) \frac{d}{d x} [x + 9]}{{(x + 9)}^{2}}$

Step 2.9

Add $22 x + 10$ and $0$ .

$\frac{(x + 9) (22 x + 10) - (11 x^{2} + 10 x - 9) \frac{d}{d x} [x + 9]}{{(x + 9)}^{2}}$

Step 2.10

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

$\frac{(x + 9) (22 x + 10) - (11 x^{2} + 10 x - 9) (\frac{d}{d x} [x] + \frac{d}{d x} [9])}{{(x + 9)}^{2}}$

Step 2.11

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

$\frac{(x + 9) (22 x + 10) - (11 x^{2} + 10 x - 9) (1 + \frac{d}{d x} [9])}{{(x + 9)}^{2}}$

Step 2.12

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

$\frac{(x + 9) (22 x + 10) - (11 x^{2} + 10 x - 9) (1 + 0)}{{(x + 9)}^{2}}$

Step 2.13

Simplify the expression.

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Step 2.13.1

Add $1$ and $0$ .

$\frac{(x + 9) (22 x + 10) - (11 x^{2} + 10 x - 9) \cdot 1}{{(x + 9)}^{2}}$

Step 2.13.2

Multiply $- 1$ by $1$ .

$\frac{(x + 9) (22 x + 10) - (11 x^{2} + 10 x - 9)}{{(x + 9)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{(x + 9) (22 x + 10) - (11 x^{2}) - (10 x) - - 9}{{(x + 9)}^{2}}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Expand $(x + 9) (22 x + 10)$ using the FOIL Method.

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Step 3.2.1.1.1

Apply the distributive property.

$\frac{x (22 x + 10) + 9 (22 x + 10) - (11 x^{2}) - (10 x) - - 9}{{(x + 9)}^{2}}$

Step 3.2.1.1.2

Apply the distributive property.

$\frac{x (22 x) + x \cdot 10 + 9 (22 x + 10) - (11 x^{2}) - (10 x) - - 9}{{(x + 9)}^{2}}$

Step 3.2.1.1.3

Apply the distributive property.

$\frac{x (22 x) + x \cdot 10 + 9 (22 x) + 9 \cdot 10 - (11 x^{2}) - (10 x) - - 9}{{(x + 9)}^{2}}$

Step 3.2.1.2

Simplify and combine like terms.

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Step 3.2.1.2.1

Simplify each term.

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Step 3.2.1.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{22 x \cdot x + x \cdot 10 + 9 (22 x) + 9 \cdot 10 - (11 x^{2}) - (10 x) - - 9}{{(x + 9)}^{2}}$

Step 3.2.1.2.1.2

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

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Step 3.2.1.2.1.2.1

Move $x$ .

$\frac{22 (x \cdot x) + x \cdot 10 + 9 (22 x) + 9 \cdot 10 - (11 x^{2}) - (10 x) - - 9}{{(x + 9)}^{2}}$

Step 3.2.1.2.1.2.2

Multiply $x$ by $x$ .

$\frac{22 x^{2} + x \cdot 10 + 9 (22 x) + 9 \cdot 10 - (11 x^{2}) - (10 x) - - 9}{{(x + 9)}^{2}}$

Step 3.2.1.2.1.3

Move $10$ to the left of $x$ .

$\frac{22 x^{2} + 10 \cdot x + 9 (22 x) + 9 \cdot 10 - (11 x^{2}) - (10 x) - - 9}{{(x + 9)}^{2}}$

Step 3.2.1.2.1.4

Multiply $22$ by $9$ .

$\frac{22 x^{2} + 10 x + 198 x + 9 \cdot 10 - (11 x^{2}) - (10 x) - - 9}{{(x + 9)}^{2}}$

Step 3.2.1.2.1.5

Multiply $9$ by $10$ .

$\frac{22 x^{2} + 10 x + 198 x + 90 - (11 x^{2}) - (10 x) - - 9}{{(x + 9)}^{2}}$

Step 3.2.1.2.2

Add $10 x$ and $198 x$ .

$\frac{22 x^{2} + 208 x + 90 - (11 x^{2}) - (10 x) - - 9}{{(x + 9)}^{2}}$

Step 3.2.1.3

Multiply $11$ by $- 1$ .

$\frac{22 x^{2} + 208 x + 90 - 11 x^{2} - (10 x) - - 9}{{(x + 9)}^{2}}$

Step 3.2.1.4

Multiply $10$ by $- 1$ .

$\frac{22 x^{2} + 208 x + 90 - 11 x^{2} - 10 x - - 9}{{(x + 9)}^{2}}$

Step 3.2.1.5

Multiply $- 1$ by $- 9$ .

$\frac{22 x^{2} + 208 x + 90 - 11 x^{2} - 10 x + 9}{{(x + 9)}^{2}}$

Step 3.2.2

Subtract $11 x^{2}$ from $22 x^{2}$ .

$\frac{11 x^{2} + 208 x + 90 - 10 x + 9}{{(x + 9)}^{2}}$

Step 3.2.3

Subtract $10 x$ from $208 x$ .

$\frac{11 x^{2} + 198 x + 90 + 9}{{(x + 9)}^{2}}$

Step 3.2.4

Add $90$ and $9$ .

$\frac{11 x^{2} + 198 x + 99}{{(x + 9)}^{2}}$

Step 3.3

Factor $11$ out of $11 x^{2} + 198 x + 99$ .

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Step 3.3.1

Factor $11$ out of $11 x^{2}$ .

$\frac{11 (x^{2}) + 198 x + 99}{{(x + 9)}^{2}}$

Step 3.3.2

Factor $11$ out of $198 x$ .

$\frac{11 (x^{2}) + 11 (18 x) + 99}{{(x + 9)}^{2}}$

Step 3.3.3

Factor $11$ out of $99$ .

$\frac{11 x^{2} + 11 (18 x) + 11 \cdot 9}{{(x + 9)}^{2}}$

Step 3.3.4

Factor $11$ out of $11 x^{2} + 11 (18 x)$ .

$\frac{11 (x^{2} + 18 x) + 11 \cdot 9}{{(x + 9)}^{2}}$

Step 3.3.5

Factor $11$ out of $11 (x^{2} + 18 x) + 11 \cdot 9$ .

$\frac{11 (x^{2} + 18 x + 9)}{{(x + 9)}^{2}}$