Calculus Examples

$y = \frac{4 x - 8}{2 x + 3}$ , $x = 2$

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 $f (x) = 4 x - 8$ and $g (x) = 2 x + 3$ .

$\frac{(2 x + 3) \frac{d}{d x} [4 x - 8] - (4 x - 8) \frac{d}{d x} [2 x + 3]}{{(2 x + 3)}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $4 x - 8$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 8]$ .

$\frac{(2 x + 3) (\frac{d}{d x} [4 x] + \frac{d}{d x} [- 8]) - (4 x - 8) \frac{d}{d x} [2 x + 3]}{{(2 x + 3)}^{2}}$

Step 2.2

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

$\frac{(2 x + 3) (4 \frac{d}{d x} [x] + \frac{d}{d x} [- 8]) - (4 x - 8) \frac{d}{d x} [2 x + 3]}{{(2 x + 3)}^{2}}$

Step 2.3

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

$\frac{(2 x + 3) (4 \cdot 1 + \frac{d}{d x} [- 8]) - (4 x - 8) \frac{d}{d x} [2 x + 3]}{{(2 x + 3)}^{2}}$

Step 2.4

Multiply $4$ by $1$ .

$\frac{(2 x + 3) (4 + \frac{d}{d x} [- 8]) - (4 x - 8) \frac{d}{d x} [2 x + 3]}{{(2 x + 3)}^{2}}$

Step 2.5

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$\frac{(2 x + 3) (4 + 0) - (4 x - 8) \frac{d}{d x} [2 x + 3]}{{(2 x + 3)}^{2}}$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $4$ and $0$ .

$\frac{(2 x + 3) \cdot 4 - (4 x - 8) \frac{d}{d x} [2 x + 3]}{{(2 x + 3)}^{2}}$

Step 2.6.2

Move $4$ to the left of $2 x + 3$ .

$\frac{4 \cdot (2 x + 3) - (4 x - 8) \frac{d}{d x} [2 x + 3]}{{(2 x + 3)}^{2}}$

Step 2.7

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

$\frac{4 (2 x + 3) - (4 x - 8) (\frac{d}{d x} [2 x] + \frac{d}{d x} [3])}{{(2 x + 3)}^{2}}$

Step 2.8

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

$\frac{4 (2 x + 3) - (4 x - 8) (2 \frac{d}{d x} [x] + \frac{d}{d x} [3])}{{(2 x + 3)}^{2}}$

Step 2.9

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

$\frac{4 (2 x + 3) - (4 x - 8) (2 \cdot 1 + \frac{d}{d x} [3])}{{(2 x + 3)}^{2}}$

Step 2.10

Multiply $2$ by $1$ .

$\frac{4 (2 x + 3) - (4 x - 8) (2 + \frac{d}{d x} [3])}{{(2 x + 3)}^{2}}$

Step 2.11

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

$\frac{4 (2 x + 3) - (4 x - 8) (2 + 0)}{{(2 x + 3)}^{2}}$

Step 2.12

Simplify the expression.

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Step 2.12.1

Add $2$ and $0$ .

$\frac{4 (2 x + 3) - (4 x - 8) \cdot 2}{{(2 x + 3)}^{2}}$

Step 2.12.2

Multiply $2$ by $- 1$ .

$\frac{4 (2 x + 3) - 2 (4 x - 8)}{{(2 x + 3)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{4 (2 x) + 4 \cdot 3 - 2 (4 x - 8)}{{(2 x + 3)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{4 (2 x) + 4 \cdot 3 - 2 (4 x) - 2 \cdot - 8}{{(2 x + 3)}^{2}}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Combine the opposite terms in $4 (2 x) + 4 \cdot 3 - 2 (4 x) - 2 \cdot - 8$ .

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Step 3.3.1.1

Reorder the factors in the terms $4 (2 x)$ and $- 2 (4 x)$ .

$\frac{2 \cdot 4 x + 4 \cdot 3 - 2 \cdot 4 x - 2 \cdot - 8}{{(2 x + 3)}^{2}}$

Step 3.3.1.2

Subtract $2 \cdot 4 x$ from $2 \cdot 4 x$ .

$\frac{4 \cdot 3 + 0 - 2 \cdot - 8}{{(2 x + 3)}^{2}}$

Step 3.3.1.3

Add $4 \cdot 3$ and $0$ .

$\frac{4 \cdot 3 - 2 \cdot - 8}{{(2 x + 3)}^{2}}$

Step 3.3.2

Simplify each term.

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Step 3.3.2.1

Multiply $4$ by $3$ .

$\frac{12 - 2 \cdot - 8}{{(2 x + 3)}^{2}}$

Step 3.3.2.2

Multiply $- 2$ by $- 8$ .

$\frac{12 + 16}{{(2 x + 3)}^{2}}$

Step 3.3.3

Add $12$ and $16$ .

$\frac{28}{{(2 x + 3)}^{2}}$

Step 4

Evaluate the derivative at $x = 2$ .

$\frac{28}{{(2 (2) + 3)}^{2}}$

Step 5

Simplify.

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Step 5.1

Simplify the denominator.

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Step 5.1.1

Multiply $2$ by $2$ .

$\frac{28}{{(4 + 3)}^{2}}$

Step 5.1.2

Add $4$ and $3$ .

$\frac{28}{7^{2}}$

Step 5.1.3

Raise $7$ to the power of $2$ .

$\frac{28}{49}$

Step 5.2

Cancel the common factor of $28$ and $49$ .

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Step 5.2.1

Factor $7$ out of $28$ .

$\frac{7 (4)}{49}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $7$ out of $49$ .

$\frac{7 \cdot 4}{7 \cdot 7}$

Step 5.2.2.2

Cancel the common factor.

$\frac{7 \cdot 4}{7 \cdot 7}$

Step 5.2.2.3

Rewrite the expression.

$\frac{4}{7}$