Algebra Examples

$\frac{2 x^{2} + 13 x + 26}{x + 4}$

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 $f (x) = 2 x^{2} + 13 x + 26$ and $g (x) = x + 4$ .

$\frac{(x + 4) \frac{d}{d x} [2 x^{2} + 13 x + 26] - (2 x^{2} + 13 x + 26) \frac{d}{d x} [x + 4]}{{(x + 4)}^{2}}$

Step 2

By the Sum Rule, the derivative of $2 x^{2} + 13 x + 26$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [13 x] + \frac{d}{d x} [26]$ .

$\frac{(x + 4) (\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [13 x] + \frac{d}{d x} [26]) - (2 x^{2} + 13 x + 26) \frac{d}{d x} [x + 4]}{{(x + 4)}^{2}}$

Step 3

Evaluate $\frac{d}{d x} [2 x^{2}]$ .

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Step 3.1

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

$\frac{(x + 4) (2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [13 x] + \frac{d}{d x} [26]) - (2 x^{2} + 13 x + 26) \frac{d}{d x} [x + 4]}{{(x + 4)}^{2}}$

Step 3.2

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

$\frac{(x + 4) (2 (2 x) + \frac{d}{d x} [13 x] + \frac{d}{d x} [26]) - (2 x^{2} + 13 x + 26) \frac{d}{d x} [x + 4]}{{(x + 4)}^{2}}$

Step 3.3

Multiply $2$ by $2$ .

$\frac{(x + 4) (4 x + \frac{d}{d x} [13 x] + \frac{d}{d x} [26]) - (2 x^{2} + 13 x + 26) \frac{d}{d x} [x + 4]}{{(x + 4)}^{2}}$

Step 4

Evaluate $\frac{d}{d x} [13 x]$ .

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Step 4.1

Since $13$ is constant with respect to $x$ , the derivative of $13 x$ with respect to $x$ is $13 \frac{d}{d x} [x]$ .

$\frac{(x + 4) (4 x + 13 \frac{d}{d x} [x] + \frac{d}{d x} [26]) - (2 x^{2} + 13 x + 26) \frac{d}{d x} [x + 4]}{{(x + 4)}^{2}}$

Step 4.2

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

$\frac{(x + 4) (4 x + 13 \cdot 1 + \frac{d}{d x} [26]) - (2 x^{2} + 13 x + 26) \frac{d}{d x} [x + 4]}{{(x + 4)}^{2}}$

Step 4.3

Multiply $13$ by $1$ .

$\frac{(x + 4) (4 x + 13 + \frac{d}{d x} [26]) - (2 x^{2} + 13 x + 26) \frac{d}{d x} [x + 4]}{{(x + 4)}^{2}}$

Step 5

Differentiate.

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Step 5.1

Since $26$ is constant with respect to $x$ , the derivative of $26$ with respect to $x$ is $0$ .

$\frac{(x + 4) (4 x + 13 + 0) - (2 x^{2} + 13 x + 26) \frac{d}{d x} [x + 4]}{{(x + 4)}^{2}}$

Step 5.2

Add $4 x + 13$ and $0$ .

$\frac{(x + 4) (4 x + 13) - (2 x^{2} + 13 x + 26) \frac{d}{d x} [x + 4]}{{(x + 4)}^{2}}$

Step 5.3

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

$\frac{(x + 4) (4 x + 13) - (2 x^{2} + 13 x + 26) (\frac{d}{d x} [x] + \frac{d}{d x} [4])}{{(x + 4)}^{2}}$

Step 5.4

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

$\frac{(x + 4) (4 x + 13) - (2 x^{2} + 13 x + 26) (1 + \frac{d}{d x} [4])}{{(x + 4)}^{2}}$

Step 5.5

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{(x + 4) (4 x + 13) - (2 x^{2} + 13 x + 26) (1 + 0)}{{(x + 4)}^{2}}$

Step 5.6

Add $1$ and $0$ .

$\frac{(x + 4) (4 x + 13) - (2 x^{2} + 13 x + 26) \cdot 1}{{(x + 4)}^{2}}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{(x + 4) (4 x + 13) + (- (2 x^{2}) - (13 x) - 1 \cdot 26) \cdot 1}{{(x + 4)}^{2}}$

Step 6.2

Apply the distributive property.

$\frac{(x + 4) (4 x + 13) - (2 x^{2}) \cdot 1 - (13 x) \cdot 1 - 1 \cdot 26 \cdot 1}{{(x + 4)}^{2}}$

Step 6.3

Simplify the numerator.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Expand $(x + 4) (4 x + 13)$ using the FOIL Method.

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Step 6.3.1.1.1

Apply the distributive property.

$\frac{x (4 x + 13) + 4 (4 x + 13) - (2 x^{2}) \cdot 1 - (13 x) \cdot 1 - 1 \cdot 26 \cdot 1}{{(x + 4)}^{2}}$

Step 6.3.1.1.2

Apply the distributive property.

$\frac{x (4 x) + x \cdot 13 + 4 (4 x + 13) - (2 x^{2}) \cdot 1 - (13 x) \cdot 1 - 1 \cdot 26 \cdot 1}{{(x + 4)}^{2}}$

Step 6.3.1.1.3

Apply the distributive property.

$\frac{x (4 x) + x \cdot 13 + 4 (4 x) + 4 \cdot 13 - (2 x^{2}) \cdot 1 - (13 x) \cdot 1 - 1 \cdot 26 \cdot 1}{{(x + 4)}^{2}}$

Step 6.3.1.2

Simplify and combine like terms.

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Step 6.3.1.2.1

Simplify each term.

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Step 6.3.1.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{4 x \cdot x + x \cdot 13 + 4 (4 x) + 4 \cdot 13 - (2 x^{2}) \cdot 1 - (13 x) \cdot 1 - 1 \cdot 26 \cdot 1}{{(x + 4)}^{2}}$

Step 6.3.1.2.1.2

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

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Step 6.3.1.2.1.2.1

Move $x$ .

$\frac{4 (x \cdot x) + x \cdot 13 + 4 (4 x) + 4 \cdot 13 - (2 x^{2}) \cdot 1 - (13 x) \cdot 1 - 1 \cdot 26 \cdot 1}{{(x + 4)}^{2}}$

Step 6.3.1.2.1.2.2

Multiply $x$ by $x$ .

$\frac{4 x^{2} + x \cdot 13 + 4 (4 x) + 4 \cdot 13 - (2 x^{2}) \cdot 1 - (13 x) \cdot 1 - 1 \cdot 26 \cdot 1}{{(x + 4)}^{2}}$

Step 6.3.1.2.1.3

Move $13$ to the left of $x$ .

$\frac{4 x^{2} + 13 \cdot x + 4 (4 x) + 4 \cdot 13 - (2 x^{2}) \cdot 1 - (13 x) \cdot 1 - 1 \cdot 26 \cdot 1}{{(x + 4)}^{2}}$

Step 6.3.1.2.1.4

Multiply $4$ by $4$ .

$\frac{4 x^{2} + 13 x + 16 x + 4 \cdot 13 - (2 x^{2}) \cdot 1 - (13 x) \cdot 1 - 1 \cdot 26 \cdot 1}{{(x + 4)}^{2}}$

Step 6.3.1.2.1.5

Multiply $4$ by $13$ .

$\frac{4 x^{2} + 13 x + 16 x + 52 - (2 x^{2}) \cdot 1 - (13 x) \cdot 1 - 1 \cdot 26 \cdot 1}{{(x + 4)}^{2}}$

Step 6.3.1.2.2

Add $13 x$ and $16 x$ .

$\frac{4 x^{2} + 29 x + 52 - (2 x^{2}) \cdot 1 - (13 x) \cdot 1 - 1 \cdot 26 \cdot 1}{{(x + 4)}^{2}}$

Step 6.3.1.3

Multiply $2$ by $- 1$ .

$\frac{4 x^{2} + 29 x + 52 - 2 x^{2} \cdot 1 - (13 x) \cdot 1 - 1 \cdot 26 \cdot 1}{{(x + 4)}^{2}}$

Step 6.3.1.4

Multiply $- 2$ by $1$ .

$\frac{4 x^{2} + 29 x + 52 - 2 x^{2} - (13 x) \cdot 1 - 1 \cdot 26 \cdot 1}{{(x + 4)}^{2}}$

Step 6.3.1.5

Multiply $13$ by $- 1$ .

$\frac{4 x^{2} + 29 x + 52 - 2 x^{2} - 13 x \cdot 1 - 1 \cdot 26 \cdot 1}{{(x + 4)}^{2}}$

Step 6.3.1.6

Multiply $- 13$ by $1$ .

$\frac{4 x^{2} + 29 x + 52 - 2 x^{2} - 13 x - 1 \cdot 26 \cdot 1}{{(x + 4)}^{2}}$

Step 6.3.1.7

Multiply $- 1 \cdot 26 \cdot 1$ .

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Step 6.3.1.7.1

Multiply $- 1$ by $26$ .

$\frac{4 x^{2} + 29 x + 52 - 2 x^{2} - 13 x - 26 \cdot 1}{{(x + 4)}^{2}}$

Step 6.3.1.7.2

Multiply $- 26$ by $1$ .

$\frac{4 x^{2} + 29 x + 52 - 2 x^{2} - 13 x - 26}{{(x + 4)}^{2}}$

Step 6.3.2

Subtract $2 x^{2}$ from $4 x^{2}$ .

$\frac{2 x^{2} + 29 x + 52 - 13 x - 26}{{(x + 4)}^{2}}$

Step 6.3.3

Subtract $13 x$ from $29 x$ .

$\frac{2 x^{2} + 16 x + 52 - 26}{{(x + 4)}^{2}}$

Step 6.3.4

Subtract $26$ from $52$ .

$\frac{2 x^{2} + 16 x + 26}{{(x + 4)}^{2}}$

Step 6.4

Factor $2$ out of $2 x^{2} + 16 x + 26$ .

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Step 6.4.1

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

$\frac{2 (x^{2}) + 16 x + 26}{{(x + 4)}^{2}}$

Step 6.4.2

Factor $2$ out of $16 x$ .

$\frac{2 (x^{2}) + 2 (8 x) + 26}{{(x + 4)}^{2}}$

Step 6.4.3

Factor $2$ out of $26$ .

$\frac{2 x^{2} + 2 (8 x) + 2 \cdot 13}{{(x + 4)}^{2}}$

Step 6.4.4

Factor $2$ out of $2 x^{2} + 2 (8 x)$ .

$\frac{2 (x^{2} + 8 x) + 2 \cdot 13}{{(x + 4)}^{2}}$

Step 6.4.5

Factor $2$ out of $2 (x^{2} + 8 x) + 2 \cdot 13$ .

$\frac{2 (x^{2} + 8 x + 13)}{{(x + 4)}^{2}}$