Calculus Examples

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

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 $f (x) = 5 x - 2$ and $g (x) = 4 x + 3$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $5 x - 2$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [- 2]$ .

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

Step 2.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

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

Step 2.3

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

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

Step 2.4

Multiply $5$ by $1$ .

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

Step 2.5

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

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

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $5$ and $0$ .

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

Step 2.6.2

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

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

Step 2.7

By the Sum Rule, the derivative of $4 x + 3$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [3]$ .

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

Multiply $4$ by $1$ .

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

Step 2.11

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

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

Step 2.12

Simplify the expression.

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Step 2.12.1

Add $4$ and $0$ .

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

Step 2.12.2

Multiply $4$ by $- 1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Simplify the numerator.

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Step 3.3.1

Combine the opposite terms in $5 (4 x) + 5 \cdot 3 - 4 (5 x) - 4 \cdot - 2$ .

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Step 3.3.1.1

Reorder the factors in the terms $5 (4 x)$ and $- 4 (5 x)$ .

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

Step 3.3.1.2

Subtract $4 \cdot 5 x$ from $4 \cdot 5 x$ .

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

Step 3.3.1.3

Add $5 \cdot 3$ and $0$ .

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

Step 3.3.2

Simplify each term.

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Step 3.3.2.1

Multiply $5$ by $3$ .

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

Step 3.3.2.2

Multiply $- 4$ by $- 2$ .

$\frac{15 + 8}{{(4 x + 3)}^{2}}$

Step 3.3.3

Add $15$ and $8$ .

$\frac{23}{{(4 x + 3)}^{2}}$