Calculus Examples

$y = \frac{4 x - 6}{3 x + 8}$ , $x = 3$

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 $f (x) = 4 x - 6$ and $g (x) = 3 x + 8$ .

$\frac{(3 x + 8) \frac{d}{d x} [4 x - 6] - (4 x - 6) \frac{d}{d x} [3 x + 8]}{{(3 x + 8)}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $4 x - 6$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 6]$ .

$\frac{(3 x + 8) (\frac{d}{d x} [4 x] + \frac{d}{d x} [- 6]) - (4 x - 6) \frac{d}{d x} [3 x + 8]}{{(3 x + 8)}^{2}}$

Step 2.2

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

$\frac{(3 x + 8) (4 \frac{d}{d x} [x] + \frac{d}{d x} [- 6]) - (4 x - 6) \frac{d}{d x} [3 x + 8]}{{(3 x + 8)}^{2}}$

Step 2.3

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

$\frac{(3 x + 8) (4 \cdot 1 + \frac{d}{d x} [- 6]) - (4 x - 6) \frac{d}{d x} [3 x + 8]}{{(3 x + 8)}^{2}}$

Step 2.4

Multiply $4$ by $1$ .

$\frac{(3 x + 8) (4 + \frac{d}{d x} [- 6]) - (4 x - 6) \frac{d}{d x} [3 x + 8]}{{(3 x + 8)}^{2}}$

Step 2.5

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$\frac{(3 x + 8) (4 + 0) - (4 x - 6) \frac{d}{d x} [3 x + 8]}{{(3 x + 8)}^{2}}$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $4$ and $0$ .

$\frac{(3 x + 8) \cdot 4 - (4 x - 6) \frac{d}{d x} [3 x + 8]}{{(3 x + 8)}^{2}}$

Step 2.6.2

Move $4$ to the left of $3 x + 8$ .

$\frac{4 \cdot (3 x + 8) - (4 x - 6) \frac{d}{d x} [3 x + 8]}{{(3 x + 8)}^{2}}$

Step 2.7

By the Sum Rule, the derivative of $3 x + 8$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [8]$ .

$\frac{4 (3 x + 8) - (4 x - 6) (\frac{d}{d x} [3 x] + \frac{d}{d x} [8])}{{(3 x + 8)}^{2}}$

Step 2.8

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

$\frac{4 (3 x + 8) - (4 x - 6) (3 \frac{d}{d x} [x] + \frac{d}{d x} [8])}{{(3 x + 8)}^{2}}$

Step 2.9

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

$\frac{4 (3 x + 8) - (4 x - 6) (3 \cdot 1 + \frac{d}{d x} [8])}{{(3 x + 8)}^{2}}$

Step 2.10

Multiply $3$ by $1$ .

$\frac{4 (3 x + 8) - (4 x - 6) (3 + \frac{d}{d x} [8])}{{(3 x + 8)}^{2}}$

Step 2.11

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

$\frac{4 (3 x + 8) - (4 x - 6) (3 + 0)}{{(3 x + 8)}^{2}}$

Step 2.12

Simplify the expression.

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Step 2.12.1

Add $3$ and $0$ .

$\frac{4 (3 x + 8) - (4 x - 6) \cdot 3}{{(3 x + 8)}^{2}}$

Step 2.12.2

Multiply $3$ by $- 1$ .

$\frac{4 (3 x + 8) - 3 (4 x - 6)}{{(3 x + 8)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{4 (3 x) + 4 \cdot 8 - 3 (4 x - 6)}{{(3 x + 8)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{4 (3 x) + 4 \cdot 8 - 3 (4 x) - 3 \cdot - 6}{{(3 x + 8)}^{2}}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Combine the opposite terms in $4 (3 x) + 4 \cdot 8 - 3 (4 x) - 3 \cdot - 6$ .

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Step 3.3.1.1

Reorder the factors in the terms $4 (3 x)$ and $- 3 (4 x)$ .

$\frac{3 \cdot 4 x + 4 \cdot 8 - 3 \cdot 4 x - 3 \cdot - 6}{{(3 x + 8)}^{2}}$

Step 3.3.1.2

Subtract $3 \cdot 4 x$ from $3 \cdot 4 x$ .

$\frac{4 \cdot 8 + 0 - 3 \cdot - 6}{{(3 x + 8)}^{2}}$

Step 3.3.1.3

Add $4 \cdot 8$ and $0$ .

$\frac{4 \cdot 8 - 3 \cdot - 6}{{(3 x + 8)}^{2}}$

Step 3.3.2

Simplify each term.

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Step 3.3.2.1

Multiply $4$ by $8$ .

$\frac{32 - 3 \cdot - 6}{{(3 x + 8)}^{2}}$

Step 3.3.2.2

Multiply $- 3$ by $- 6$ .

$\frac{32 + 18}{{(3 x + 8)}^{2}}$

Step 3.3.3

Add $32$ and $18$ .

$\frac{50}{{(3 x + 8)}^{2}}$

Step 4

Evaluate the derivative at $x = 3$ .

$\frac{50}{{(3 (3) + 8)}^{2}}$

Step 5

Simplify the denominator.

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Step 5.1

Multiply $3$ by $3$ .

$\frac{50}{{(9 + 8)}^{2}}$

Step 5.2

Add $9$ and $8$ .

$\frac{50}{17^{2}}$

Step 5.3

Raise $17$ to the power of $2$ .

$\frac{50}{289}$