Calculus Examples

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

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{x^{2} - 4}{5 x - 3})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $f (x) = x^{2} - 4$ and $g (x) = 5 x - 3$ .

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

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $x^{2} - 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4]$ .

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

Step 3.2.2

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

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

Step 3.2.3

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

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

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

Add $2 x$ and $0$ .

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

Step 3.2.4.2

Move $2$ to the left of $5 x - 3$ .

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

Step 3.2.5

By the Sum Rule, the derivative of $5 x - 3$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [- 3]$ .

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

Step 3.2.6

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

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

Step 3.2.7

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

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

Step 3.2.8

Multiply $5$ by $1$ .

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

Step 3.2.9

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

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

Step 3.2.10

Simplify the expression.

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Step 3.2.10.1

Add $5$ and $0$ .

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

Step 3.2.10.2

Multiply $5$ by $- 1$ .

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

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

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

Step 3.3.2

Apply the distributive property.

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

Step 3.3.3

Apply the distributive property.

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

Step 3.3.4

Simplify the numerator.

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Step 3.3.4.1

Simplify each term.

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Step 3.3.4.1.1

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

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Step 3.3.4.1.1.1

Move $x$ .

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

Step 3.3.4.1.1.2

Multiply $x$ by $x$ .

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

Step 3.3.4.1.2

Multiply $5$ by $2$ .

$\frac{10 x^{2} + 2 \cdot - 3 x - 5 x^{2} - 5 \cdot - 4}{{(5 x - 3)}^{2}}$

Step 3.3.4.1.3

Multiply $2$ by $- 3$ .

$\frac{10 x^{2} - 6 x - 5 x^{2} - 5 \cdot - 4}{{(5 x - 3)}^{2}}$

Step 3.3.4.1.4

Multiply $- 5$ by $- 4$ .

$\frac{10 x^{2} - 6 x - 5 x^{2} + 20}{{(5 x - 3)}^{2}}$

Step 3.3.4.2

Subtract $5 x^{2}$ from $10 x^{2}$ .

$\frac{5 x^{2} - 6 x + 20}{{(5 x - 3)}^{2}}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \frac{5 x^{2} - 6 x + 20}{{(5 x - 3)}^{2}}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{5 x^{2} - 6 x + 20}{{(5 x - 3)}^{2}}$