Calculus Examples

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

Step 1

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{3 x^{2} - x - 2}{3 x^{2}}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [\frac{3 x^{2} - x - 2}{x^{2}}]$ .

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

Step 2

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

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

Step 3

Differentiate.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Multiply $2$ by $2$ .

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

Step 3.2

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

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

Step 3.3

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

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

Step 3.4

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

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

Step 3.5

Multiply $2$ by $3$ .

$\frac{1}{3} \cdot \frac{x^{2} (6 x + \frac{d}{d x} [- x] + \frac{d}{d x} [- 2]) - (3 x^{2} - x - 2) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 3.6

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$\frac{1}{3} \cdot \frac{x^{2} (6 x - \frac{d}{d x} [x] + \frac{d}{d x} [- 2]) - (3 x^{2} - x - 2) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 3.7

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

$\frac{1}{3} \cdot \frac{x^{2} (6 x - 1 \cdot 1 + \frac{d}{d x} [- 2]) - (3 x^{2} - x - 2) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 3.8

Multiply $- 1$ by $1$ .

$\frac{1}{3} \cdot \frac{x^{2} (6 x - 1 + \frac{d}{d x} [- 2]) - (3 x^{2} - x - 2) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 3.9

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

$\frac{1}{3} \cdot \frac{x^{2} (6 x - 1 + 0) - (3 x^{2} - x - 2) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 3.10

Add $6 x - 1$ and $0$ .

$\frac{1}{3} \cdot \frac{x^{2} (6 x - 1) - (3 x^{2} - x - 2) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 3.11

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

$\frac{1}{3} \cdot \frac{x^{2} (6 x - 1) - (3 x^{2} - x - 2) (2 x)}{x^{4}}$

Step 3.12

Simplify with factoring out.

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Step 3.12.1

Multiply $2$ by $- 1$ .

$\frac{1}{3} \cdot \frac{x^{2} (6 x - 1) - 2 (3 x^{2} - x - 2) x}{x^{4}}$

Step 3.12.2

Factor $x$ out of $x^{2} (6 x - 1) - 2 (3 x^{2} - x - 2) x$ .

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Step 3.12.2.1

Factor $x$ out of $x^{2} (6 x - 1)$ .

$\frac{1}{3} \cdot \frac{x (x (6 x - 1)) - 2 (3 x^{2} - x - 2) x}{x^{4}}$

Step 3.12.2.2

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

$\frac{1}{3} \cdot \frac{x (x (6 x - 1)) + x (- 2 (3 x^{2} - x - 2))}{x^{4}}$

Step 3.12.2.3

Factor $x$ out of $x (x (6 x - 1)) + x (- 2 (3 x^{2} - x - 2))$ .

$\frac{1}{3} \cdot \frac{x (x (6 x - 1) - 2 (3 x^{2} - x - 2))}{x^{4}}$

Step 4

Cancel the common factors.

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Step 4.1

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

$\frac{1}{3} \cdot \frac{x (x (6 x - 1) - 2 (3 x^{2} - x - 2))}{x \cdot x^{3}}$

Step 4.2

Cancel the common factor.

$\frac{1}{3} \cdot \frac{x (x (6 x - 1) - 2 (3 x^{2} - x - 2))}{x \cdot x^{3}}$

Step 4.3

Rewrite the expression.

$\frac{1}{3} \cdot \frac{x (6 x - 1) - 2 (3 x^{2} - x - 2)}{x^{3}}$

Step 5

Multiply $\frac{1}{3}$ by $\frac{x (6 x - 1) - 2 (3 x^{2} - x - 2)}{x^{3}}$ .

$\frac{x (6 x - 1) - 2 (3 x^{2} - x - 2)}{3 x^{3}}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{x (6 x) + x \cdot - 1 - 2 (3 x^{2} - x - 2)}{3 x^{3}}$

Step 6.2

Apply the distributive property.

$\frac{x (6 x) + x \cdot - 1 - 2 (3 x^{2}) - 2 (- x) - 2 \cdot - 2}{3 x^{3}}$

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.

$\frac{6 x \cdot x + x \cdot - 1 - 2 (3 x^{2}) - 2 (- x) - 2 \cdot - 2}{3 x^{3}}$

Step 6.3.1.2

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

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Step 6.3.1.2.1

Move $x$ .

$\frac{6 (x \cdot x) + x \cdot - 1 - 2 (3 x^{2}) - 2 (- x) - 2 \cdot - 2}{3 x^{3}}$

Step 6.3.1.2.2

Multiply $x$ by $x$ .

$\frac{6 x^{2} + x \cdot - 1 - 2 (3 x^{2}) - 2 (- x) - 2 \cdot - 2}{3 x^{3}}$

Step 6.3.1.3

Move $- 1$ to the left of $x$ .

$\frac{6 x^{2} - 1 \cdot x - 2 (3 x^{2}) - 2 (- x) - 2 \cdot - 2}{3 x^{3}}$

Step 6.3.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 6.3.1.5

Multiply $3$ by $- 2$ .

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

Step 6.3.1.6

Multiply $- 1$ by $- 2$ .

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

Step 6.3.1.7

Multiply $- 2$ by $- 2$ .

$\frac{6 x^{2} - x - 6 x^{2} + 2 x + 4}{3 x^{3}}$

Step 6.3.2

Combine the opposite terms in $6 x^{2} - x - 6 x^{2} + 2 x + 4$ .

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Step 6.3.2.1

Subtract $6 x^{2}$ from $6 x^{2}$ .

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

Step 6.3.2.2

Add $- x$ and $0$ .

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

Step 6.3.3

Add $- x$ and $2 x$ .

$\frac{x + 4}{3 x^{3}}$