Precalculus Examples

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

Step 1

Factor $x$ out of $5 x^{4} + 3 x^{2} - x$ .

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Step 1.1

Factor $x$ out of $5 x^{4}$ .

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

Step 1.2

Factor $x$ out of $3 x^{2}$ .

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

Step 1.3

Factor $x$ out of $- x$ .

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

Step 1.4

Factor $x$ out of $x (5 x^{3}) + x (3 x)$ .

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

Step 1.5

Factor $x$ out of $x (5 x^{3} + 3 x) + x \cdot - 1$ .

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

Step 2

Cancel the common factors.

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Step 2.1

Factor $x$ out of $x^{3}$ .

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

Step 2.2

Cancel the common factor.

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

Step 2.3

Rewrite the expression.

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

Step 3

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 $f (x) = 5 x^{3} + 3 x - 1$ and $g (x) = x^{2}$ .

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

Step 4

Differentiate.

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Step 4.1

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 4.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.1.2

Multiply $2$ by $2$ .

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

Step 4.2

By the Sum Rule, the derivative of $5 x^{3} + 3 x - 1$ with respect to $x$ is $\frac{d}{d x} [5 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]$ .

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

Step 4.3

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{3}$ with respect to $x$ is $5 \frac{d}{d x} [x^{3}]$ .

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

Step 4.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 4.5

Multiply $3$ by $5$ .

$\frac{x^{2} (15 x^{2} + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]) - (5 x^{3} + 3 x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4.6

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

$\frac{x^{2} (15 x^{2} + 3 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]) - (5 x^{3} + 3 x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4.7

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

$\frac{x^{2} (15 x^{2} + 3 \cdot 1 + \frac{d}{d x} [- 1]) - (5 x^{3} + 3 x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4.8

Multiply $3$ by $1$ .

$\frac{x^{2} (15 x^{2} + 3 + \frac{d}{d x} [- 1]) - (5 x^{3} + 3 x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4.9

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$\frac{x^{2} (15 x^{2} + 3 + 0) - (5 x^{3} + 3 x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4.10

Add $15 x^{2} + 3$ and $0$ .

$\frac{x^{2} (15 x^{2} + 3) - (5 x^{3} + 3 x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4.11

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

$\frac{x^{2} (15 x^{2} + 3) - (5 x^{3} + 3 x - 1) (2 x)}{x^{4}}$

Step 4.12

Simplify with factoring out.

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Step 4.12.1

Multiply $2$ by $- 1$ .

$\frac{x^{2} (15 x^{2} + 3) - 2 (5 x^{3} + 3 x - 1) x}{x^{4}}$

Step 4.12.2

Factor $x$ out of $x^{2} (15 x^{2} + 3) - 2 (5 x^{3} + 3 x - 1) x$ .

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Step 4.12.2.1

Factor $x$ out of $x^{2} (15 x^{2} + 3)$ .

$\frac{x (x (15 x^{2} + 3)) - 2 (5 x^{3} + 3 x - 1) x}{x^{4}}$

Step 4.12.2.2

Factor $x$ out of $- 2 (5 x^{3} + 3 x - 1) x$ .

$\frac{x (x (15 x^{2} + 3)) + x (- 2 (5 x^{3} + 3 x - 1))}{x^{4}}$

Step 4.12.2.3

Factor $x$ out of $x (x (15 x^{2} + 3)) + x (- 2 (5 x^{3} + 3 x - 1))$ .

$\frac{x (x (15 x^{2} + 3) - 2 (5 x^{3} + 3 x - 1))}{x^{4}}$

Step 5

Cancel the common factors.

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Step 5.1

Factor $x$ out of $x^{4}$ .

$\frac{x (x (15 x^{2} + 3) - 2 (5 x^{3} + 3 x - 1))}{x \cdot x^{3}}$

Step 5.2

Cancel the common factor.

$\frac{x (x (15 x^{2} + 3) - 2 (5 x^{3} + 3 x - 1))}{x \cdot x^{3}}$

Step 5.3

Rewrite the expression.

$\frac{x (15 x^{2} + 3) - 2 (5 x^{3} + 3 x - 1)}{x^{3}}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{x (15 x^{2}) + x \cdot 3 - 2 (5 x^{3} + 3 x - 1)}{x^{3}}$

Step 6.2

Apply the distributive property.

$\frac{x (15 x^{2}) + x \cdot 3 - 2 (5 x^{3}) - 2 (3 x) - 2 \cdot - 1}{x^{3}}$

Step 6.3

Simplify the numerator.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{15 x \cdot x^{2} + x \cdot 3 - 2 (5 x^{3}) - 2 (3 x) - 2 \cdot - 1}{x^{3}}$

Step 6.3.1.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 6.3.1.2.1

Move $x^{2}$ .

$\frac{15 (x^{2} x) + x \cdot 3 - 2 (5 x^{3}) - 2 (3 x) - 2 \cdot - 1}{x^{3}}$

Step 6.3.1.2.2

Multiply $x^{2}$ by $x$ .

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Step 6.3.1.2.2.1

Raise $x$ to the power of $1$ .

$\frac{15 (x^{2} x^{1}) + x \cdot 3 - 2 (5 x^{3}) - 2 (3 x) - 2 \cdot - 1}{x^{3}}$

Step 6.3.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{15 x^{2 + 1} + x \cdot 3 - 2 (5 x^{3}) - 2 (3 x) - 2 \cdot - 1}{x^{3}}$

Step 6.3.1.2.3

Add $2$ and $1$ .

$\frac{15 x^{3} + x \cdot 3 - 2 (5 x^{3}) - 2 (3 x) - 2 \cdot - 1}{x^{3}}$

Step 6.3.1.3

Move $3$ to the left of $x$ .

$\frac{15 x^{3} + 3 \cdot x - 2 (5 x^{3}) - 2 (3 x) - 2 \cdot - 1}{x^{3}}$

Step 6.3.1.4

Multiply $5$ by $- 2$ .

$\frac{15 x^{3} + 3 x - 10 x^{3} - 2 (3 x) - 2 \cdot - 1}{x^{3}}$

Step 6.3.1.5

Multiply $3$ by $- 2$ .

$\frac{15 x^{3} + 3 x - 10 x^{3} - 6 x - 2 \cdot - 1}{x^{3}}$

Step 6.3.1.6

Multiply $- 2$ by $- 1$ .

$\frac{15 x^{3} + 3 x - 10 x^{3} - 6 x + 2}{x^{3}}$

Step 6.3.2

Subtract $10 x^{3}$ from $15 x^{3}$ .

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

Step 6.3.3

Subtract $6 x$ from $3 x$ .

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