Calculus Examples

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

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Multiply $2 - 5 x^{2}$ by $1$ .

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Add $0$ and $\frac{d}{d x} [- 5 x^{2}]$ .

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

Step 2.9

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{2}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{2}]$ .

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

Step 2.10

Multiply $- 5$ by $- 1$ .

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

Step 2.11

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

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

Step 2.12

Multiply $2$ by $5$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Simplify the numerator.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $10$ by $3$ .

$\frac{2 - 5 x^{2} + 30 x + 10 x \cdot x}{{(2 - 5 x^{2})}^{2}}$

Step 3.3.1.2

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

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Step 3.3.1.2.1

Move $x$ .

$\frac{2 - 5 x^{2} + 30 x + 10 (x \cdot x)}{{(2 - 5 x^{2})}^{2}}$

Step 3.3.1.2.2

Multiply $x$ by $x$ .

$\frac{2 - 5 x^{2} + 30 x + 10 x^{2}}{{(2 - 5 x^{2})}^{2}}$

Step 3.3.2

Add $- 5 x^{2}$ and $10 x^{2}$ .

$\frac{2 + 5 x^{2} + 30 x}{{(2 - 5 x^{2})}^{2}}$

Step 3.4

Reorder terms.

$\frac{5 x^{2} + 30 x + 2}{{(- 5 x^{2} + 2)}^{2}}$