Calculus Examples

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

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 $f (x) = x^{3} - 6$ and $g (x) = x^{2}$ .

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

Step 2

Differentiate.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Multiply $2$ by $2$ .

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

Step 2.2

By the Sum Rule, the derivative of $x^{3} - 6$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 6]$ .

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

Step 2.3

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

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

Step 2.4

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

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

Step 2.5

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

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

Step 3

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

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Step 3.1

Move $x^{2}$ .

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

Step 3.2

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

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

Step 3.3

Add $2$ and $2$ .

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

Step 4

Move $3$ to the left of $x^{4}$ .

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

Step 5

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

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

Step 6

Simplify with factoring out.

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Step 6.1

Multiply $2$ by $- 1$ .

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

Step 6.2

Factor $x$ out of $3 x^{4} - 2 (x^{3} - 6) x$ .

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Step 6.2.1

Factor $x$ out of $3 x^{4}$ .

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

Step 6.2.2

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

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

Step 6.2.3

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

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

Step 7

Cancel the common factors.

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Step 7.1

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

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

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

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

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

Step 8.2

Simplify the numerator.

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Step 8.2.1

Multiply $- 2$ by $- 6$ .

$\frac{3 x^{3} - 2 x^{3} + 12}{x^{3}}$

Step 8.2.2

Subtract $2 x^{3}$ from $3 x^{3}$ .

$\frac{x^{3} + 12}{x^{3}}$