Calculus Examples

$k (x) = \frac{9 x + 3}{7 x^{2} + x}$

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 $f (x) = 9 x + 3$ and $g (x) = 7 x^{2} + x$ .

$\frac{(7 x^{2} + x) \frac{d}{d x} [9 x + 3] - (9 x + 3) \frac{d}{d x} [7 x^{2} + x]}{{(7 x^{2} + x)}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $9 x + 3$ with respect to $x$ is $\frac{d}{d x} [9 x] + \frac{d}{d x} [3]$ .

$\frac{(7 x^{2} + x) (\frac{d}{d x} [9 x] + \frac{d}{d x} [3]) - (9 x + 3) \frac{d}{d x} [7 x^{2} + x]}{{(7 x^{2} + x)}^{2}}$

Step 2.2

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$\frac{(7 x^{2} + x) (9 \frac{d}{d x} [x] + \frac{d}{d x} [3]) - (9 x + 3) \frac{d}{d x} [7 x^{2} + x]}{{(7 x^{2} + x)}^{2}}$

Step 2.3

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

$\frac{(7 x^{2} + x) (9 \cdot 1 + \frac{d}{d x} [3]) - (9 x + 3) \frac{d}{d x} [7 x^{2} + x]}{{(7 x^{2} + x)}^{2}}$

Step 2.4

Multiply $9$ by $1$ .

$\frac{(7 x^{2} + x) (9 + \frac{d}{d x} [3]) - (9 x + 3) \frac{d}{d x} [7 x^{2} + x]}{{(7 x^{2} + x)}^{2}}$

Step 2.5

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

$\frac{(7 x^{2} + x) (9 + 0) - (9 x + 3) \frac{d}{d x} [7 x^{2} + x]}{{(7 x^{2} + x)}^{2}}$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $9$ and $0$ .

$\frac{(7 x^{2} + x) \cdot 9 - (9 x + 3) \frac{d}{d x} [7 x^{2} + x]}{{(7 x^{2} + x)}^{2}}$

Step 2.6.2

Move $9$ to the left of $7 x^{2} + x$ .

$\frac{9 \cdot (7 x^{2} + x) - (9 x + 3) \frac{d}{d x} [7 x^{2} + x]}{{(7 x^{2} + x)}^{2}}$

Step 2.7

By the Sum Rule, the derivative of $7 x^{2} + x$ with respect to $x$ is $\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [x]$ .

$\frac{9 (7 x^{2} + x) - (9 x + 3) (\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [x])}{{(7 x^{2} + x)}^{2}}$

Step 2.8

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{2}$ with respect to $x$ is $7 \frac{d}{d x} [x^{2}]$ .

$\frac{9 (7 x^{2} + x) - (9 x + 3) (7 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [x])}{{(7 x^{2} + x)}^{2}}$

Step 2.9

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

$\frac{9 (7 x^{2} + x) - (9 x + 3) (7 (2 x) + \frac{d}{d x} [x])}{{(7 x^{2} + x)}^{2}}$

Step 2.10

Multiply $2$ by $7$ .

$\frac{9 (7 x^{2} + x) - (9 x + 3) (14 x + \frac{d}{d x} [x])}{{(7 x^{2} + x)}^{2}}$

Step 2.11

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

$\frac{9 (7 x^{2} + x) - (9 x + 3) (14 x + 1)}{{(7 x^{2} + x)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{9 (7 x^{2}) + 9 x - (9 x + 3) (14 x + 1)}{{(7 x^{2} + x)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{9 (7 x^{2}) + 9 x + (- (9 x) - 1 \cdot 3) (14 x + 1)}{{(7 x^{2} + x)}^{2}}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $7$ by $9$ .

$\frac{63 x^{2} + 9 x + (- (9 x) - 1 \cdot 3) (14 x + 1)}{{(7 x^{2} + x)}^{2}}$

Step 3.3.1.2

Simplify each term.

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Step 3.3.1.2.1

Multiply $9$ by $- 1$ .

$\frac{63 x^{2} + 9 x + (- 9 x - 1 \cdot 3) (14 x + 1)}{{(7 x^{2} + x)}^{2}}$

Step 3.3.1.2.2

Multiply $- 1$ by $3$ .

$\frac{63 x^{2} + 9 x + (- 9 x - 3) (14 x + 1)}{{(7 x^{2} + x)}^{2}}$

Step 3.3.1.3

Expand $(- 9 x - 3) (14 x + 1)$ using the FOIL Method.

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Step 3.3.1.3.1

Apply the distributive property.

$\frac{63 x^{2} + 9 x - 9 x (14 x + 1) - 3 (14 x + 1)}{{(7 x^{2} + x)}^{2}}$

Step 3.3.1.3.2

Apply the distributive property.

$\frac{63 x^{2} + 9 x - 9 x (14 x) - 9 x \cdot 1 - 3 (14 x + 1)}{{(7 x^{2} + x)}^{2}}$

Step 3.3.1.3.3

Apply the distributive property.

$\frac{63 x^{2} + 9 x - 9 x (14 x) - 9 x \cdot 1 - 3 (14 x) - 3 \cdot 1}{{(7 x^{2} + x)}^{2}}$

Step 3.3.1.4

Simplify and combine like terms.

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Step 3.3.1.4.1

Simplify each term.

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Step 3.3.1.4.1.1

Rewrite using the commutative property of multiplication.

$\frac{63 x^{2} + 9 x - 9 \cdot 14 x \cdot x - 9 x \cdot 1 - 3 (14 x) - 3 \cdot 1}{{(7 x^{2} + x)}^{2}}$

Step 3.3.1.4.1.2

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

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Step 3.3.1.4.1.2.1

Move $x$ .

$\frac{63 x^{2} + 9 x - 9 \cdot 14 (x \cdot x) - 9 x \cdot 1 - 3 (14 x) - 3 \cdot 1}{{(7 x^{2} + x)}^{2}}$

Step 3.3.1.4.1.2.2

Multiply $x$ by $x$ .

$\frac{63 x^{2} + 9 x - 9 \cdot 14 x^{2} - 9 x \cdot 1 - 3 (14 x) - 3 \cdot 1}{{(7 x^{2} + x)}^{2}}$

Step 3.3.1.4.1.3

Multiply $- 9$ by $14$ .

$\frac{63 x^{2} + 9 x - 126 x^{2} - 9 x \cdot 1 - 3 (14 x) - 3 \cdot 1}{{(7 x^{2} + x)}^{2}}$

Step 3.3.1.4.1.4

Multiply $- 9$ by $1$ .

$\frac{63 x^{2} + 9 x - 126 x^{2} - 9 x - 3 (14 x) - 3 \cdot 1}{{(7 x^{2} + x)}^{2}}$

Step 3.3.1.4.1.5

Multiply $14$ by $- 3$ .

$\frac{63 x^{2} + 9 x - 126 x^{2} - 9 x - 42 x - 3 \cdot 1}{{(7 x^{2} + x)}^{2}}$

Step 3.3.1.4.1.6

Multiply $- 3$ by $1$ .

$\frac{63 x^{2} + 9 x - 126 x^{2} - 9 x - 42 x - 3}{{(7 x^{2} + x)}^{2}}$

Step 3.3.1.4.2

Subtract $42 x$ from $- 9 x$ .

$\frac{63 x^{2} + 9 x - 126 x^{2} - 51 x - 3}{{(7 x^{2} + x)}^{2}}$

Step 3.3.2

Subtract $126 x^{2}$ from $63 x^{2}$ .

$\frac{- 63 x^{2} + 9 x - 51 x - 3}{{(7 x^{2} + x)}^{2}}$

Step 3.3.3

Subtract $51 x$ from $9 x$ .

$\frac{- 63 x^{2} - 42 x - 3}{{(7 x^{2} + x)}^{2}}$

Step 3.4

Factor $3$ out of $- 63 x^{2} - 42 x - 3$ .

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Step 3.4.1

Factor $3$ out of $- 63 x^{2}$ .

$\frac{3 (- 21 x^{2}) - 42 x - 3}{{(7 x^{2} + x)}^{2}}$

Step 3.4.2

Factor $3$ out of $- 42 x$ .

$\frac{3 (- 21 x^{2}) + 3 (- 14 x) - 3}{{(7 x^{2} + x)}^{2}}$

Step 3.4.3

Factor $3$ out of $- 3$ .

$\frac{3 (- 21 x^{2}) + 3 (- 14 x) + 3 (- 1)}{{(7 x^{2} + x)}^{2}}$

Step 3.4.4

Factor $3$ out of $3 (- 21 x^{2}) + 3 (- 14 x)$ .

$\frac{3 (- 21 x^{2} - 14 x) + 3 (- 1)}{{(7 x^{2} + x)}^{2}}$

Step 3.4.5

Factor $3$ out of $3 (- 21 x^{2} - 14 x) + 3 (- 1)$ .

$\frac{3 (- 21 x^{2} - 14 x - 1)}{{(7 x^{2} + x)}^{2}}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Factor $x$ out of $7 x^{2} + x$ .

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Step 3.5.1.1

Factor $x$ out of $7 x^{2}$ .

$\frac{3 (- 21 x^{2} - 14 x - 1)}{{(x (7 x) + x)}^{2}}$

Step 3.5.1.2

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

$\frac{3 (- 21 x^{2} - 14 x - 1)}{{(x (7 x) + x^{1})}^{2}}$

Step 3.5.1.3

Factor $x$ out of $x^{1}$ .

$\frac{3 (- 21 x^{2} - 14 x - 1)}{{(x (7 x) + x \cdot 1)}^{2}}$

Step 3.5.1.4

Factor $x$ out of $x (7 x) + x \cdot 1$ .

$\frac{3 (- 21 x^{2} - 14 x - 1)}{{(x (7 x + 1))}^{2}}$

Step 3.5.2

Apply the product rule to $x (7 x + 1)$ .

$\frac{3 (- 21 x^{2} - 14 x - 1)}{x^{2} {(7 x + 1)}^{2}}$

Step 3.6

Factor $- 1$ out of $- 21 x^{2}$ .

$\frac{3 (- (21 x^{2}) - 14 x - 1)}{x^{2} {(7 x + 1)}^{2}}$

Step 3.7

Factor $- 1$ out of $- 14 x$ .

$\frac{3 (- (21 x^{2}) - (14 x) - 1)}{x^{2} {(7 x + 1)}^{2}}$

Step 3.8

Factor $- 1$ out of $- (21 x^{2}) - (14 x)$ .

$\frac{3 (- (21 x^{2} + 14 x) - 1)}{x^{2} {(7 x + 1)}^{2}}$

Step 3.9

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{3 (- (21 x^{2} + 14 x) - 1 (1))}{x^{2} {(7 x + 1)}^{2}}$

Step 3.10

Factor $- 1$ out of $- (21 x^{2} + 14 x) - 1 (1)$ .

$\frac{3 (- (21 x^{2} + 14 x + 1))}{x^{2} {(7 x + 1)}^{2}}$

Step 3.11

Rewrite $- (21 x^{2} + 14 x + 1)$ as $- 1 (21 x^{2} + 14 x + 1)$ .

$\frac{3 (- 1 (21 x^{2} + 14 x + 1))}{x^{2} {(7 x + 1)}^{2}}$

Step 3.12

Move the negative in front of the fraction.

$- \frac{3 (21 x^{2} + 14 x + 1)}{x^{2} {(7 x + 1)}^{2}}$