Calculus Examples

$\frac{3 - 5 x}{2 x^{2} - 3}$

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Add $0$ and $\frac{d}{d x} [- 5 x]$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Simplify the expression.

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Step 2.6.1

Multiply $- 5$ by $1$ .

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

Step 2.6.2

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

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

Multiply $2$ by $2$ .

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

Step 2.11

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

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

Step 2.12

Simplify the expression.

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Step 2.12.1

Add $4 x$ and $0$ .

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

Step 2.12.2

Multiply $4$ by $- 1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 3.4

Simplify the numerator.

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Step 3.4.1

Simplify each term.

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Step 3.4.1.1

Multiply $2$ by $- 5$ .

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

Step 3.4.1.2

Multiply $- 5$ by $- 3$ .

$\frac{- 10 x^{2} + 15 - 4 \cdot 3 x - 4 (- 5 x) x}{{(2 x^{2} - 3)}^{2}}$

Step 3.4.1.3

Multiply $- 4$ by $3$ .

$\frac{- 10 x^{2} + 15 - 12 x - 4 (- 5 x) x}{{(2 x^{2} - 3)}^{2}}$

Step 3.4.1.4

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

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Step 3.4.1.4.1

Move $x$ .

$\frac{- 10 x^{2} + 15 - 12 x - 4 (- 5 (x \cdot x))}{{(2 x^{2} - 3)}^{2}}$

Step 3.4.1.4.2

Multiply $x$ by $x$ .

$\frac{- 10 x^{2} + 15 - 12 x - 4 (- 5 x^{2})}{{(2 x^{2} - 3)}^{2}}$

Step 3.4.1.5

Multiply $- 5$ by $- 4$ .

$\frac{- 10 x^{2} + 15 - 12 x + 20 x^{2}}{{(2 x^{2} - 3)}^{2}}$

Step 3.4.2

Add $- 10 x^{2}$ and $20 x^{2}$ .

$\frac{10 x^{2} + 15 - 12 x}{{(2 x^{2} - 3)}^{2}}$

Step 3.5

Reorder terms.

$\frac{10 x^{2} - 12 x + 15}{{(2 x^{2} - 3)}^{2}}$