Calculus Examples

$y = \frac{3 x - 9}{4 x + 5}$ , $x = 2$

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 $f (x) = 3 x - 9$ and $g (x) = 4 x + 5$ .

$\frac{(4 x + 5) \frac{d}{d x} [3 x - 9] - (3 x - 9) \frac{d}{d x} [4 x + 5]}{{(4 x + 5)}^{2}}$

Step 2

Differentiate.

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Step 2.1

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

$\frac{(4 x + 5) (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 9]) - (3 x - 9) \frac{d}{d x} [4 x + 5]}{{(4 x + 5)}^{2}}$

Step 2.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$\frac{(4 x + 5) (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 9]) - (3 x - 9) \frac{d}{d x} [4 x + 5]}{{(4 x + 5)}^{2}}$

Step 2.3

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

$\frac{(4 x + 5) (3 \cdot 1 + \frac{d}{d x} [- 9]) - (3 x - 9) \frac{d}{d x} [4 x + 5]}{{(4 x + 5)}^{2}}$

Step 2.4

Multiply $3$ by $1$ .

$\frac{(4 x + 5) (3 + \frac{d}{d x} [- 9]) - (3 x - 9) \frac{d}{d x} [4 x + 5]}{{(4 x + 5)}^{2}}$

Step 2.5

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

$\frac{(4 x + 5) (3 + 0) - (3 x - 9) \frac{d}{d x} [4 x + 5]}{{(4 x + 5)}^{2}}$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $3$ and $0$ .

$\frac{(4 x + 5) \cdot 3 - (3 x - 9) \frac{d}{d x} [4 x + 5]}{{(4 x + 5)}^{2}}$

Step 2.6.2

Move $3$ to the left of $4 x + 5$ .

$\frac{3 \cdot (4 x + 5) - (3 x - 9) \frac{d}{d x} [4 x + 5]}{{(4 x + 5)}^{2}}$

Step 2.7

By the Sum Rule, the derivative of $4 x + 5$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [5]$ .

$\frac{3 (4 x + 5) - (3 x - 9) (\frac{d}{d x} [4 x] + \frac{d}{d x} [5])}{{(4 x + 5)}^{2}}$

Step 2.8

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$\frac{3 (4 x + 5) - (3 x - 9) (4 \frac{d}{d x} [x] + \frac{d}{d x} [5])}{{(4 x + 5)}^{2}}$

Step 2.9

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

$\frac{3 (4 x + 5) - (3 x - 9) (4 \cdot 1 + \frac{d}{d x} [5])}{{(4 x + 5)}^{2}}$

Step 2.10

Multiply $4$ by $1$ .

$\frac{3 (4 x + 5) - (3 x - 9) (4 + \frac{d}{d x} [5])}{{(4 x + 5)}^{2}}$

Step 2.11

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{3 (4 x + 5) - (3 x - 9) (4 + 0)}{{(4 x + 5)}^{2}}$

Step 2.12

Simplify the expression.

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Step 2.12.1

Add $4$ and $0$ .

$\frac{3 (4 x + 5) - (3 x - 9) \cdot 4}{{(4 x + 5)}^{2}}$

Step 2.12.2

Multiply $4$ by $- 1$ .

$\frac{3 (4 x + 5) - 4 (3 x - 9)}{{(4 x + 5)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{3 (4 x) + 3 \cdot 5 - 4 (3 x - 9)}{{(4 x + 5)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{3 (4 x) + 3 \cdot 5 - 4 (3 x) - 4 \cdot - 9}{{(4 x + 5)}^{2}}$

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 $4$ by $3$ .

$\frac{12 x + 3 \cdot 5 - 4 (3 x) - 4 \cdot - 9}{{(4 x + 5)}^{2}}$

Step 3.3.1.2

Multiply $3$ by $5$ .

$\frac{12 x + 15 - 4 (3 x) - 4 \cdot - 9}{{(4 x + 5)}^{2}}$

Step 3.3.1.3

Multiply $3$ by $- 4$ .

$\frac{12 x + 15 - 12 x - 4 \cdot - 9}{{(4 x + 5)}^{2}}$

Step 3.3.1.4

Multiply $- 4$ by $- 9$ .

$\frac{12 x + 15 - 12 x + 36}{{(4 x + 5)}^{2}}$

Step 3.3.2

Combine the opposite terms in $12 x + 15 - 12 x + 36$ .

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Step 3.3.2.1

Subtract $12 x$ from $12 x$ .

$\frac{0 + 15 + 36}{{(4 x + 5)}^{2}}$

Step 3.3.2.2

Add $0$ and $15$ .

$\frac{15 + 36}{{(4 x + 5)}^{2}}$

Step 3.3.3

Add $15$ and $36$ .

$\frac{51}{{(4 x + 5)}^{2}}$

Step 4

Evaluate the derivative at $x = 2$ .

$\frac{51}{{(4 (2) + 5)}^{2}}$

Step 5

Simplify the denominator.

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Step 5.1

Multiply $4$ by $2$ .

$\frac{51}{{(8 + 5)}^{2}}$

Step 5.2

Add $8$ and $5$ .

$\frac{51}{13^{2}}$

Step 5.3

Raise $13$ to the power of $2$ .

$\frac{51}{169}$