Calculus Examples

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

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 $f (x) = x^{3} - 3 x^{2} + 4$ and $g (x) = x^{2}$ .

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.1.2

Multiply $2$ by $2$ .

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

Step 2.2

By the Sum Rule, the derivative of $x^{3} - 3 x^{2} + 4$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [4]$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Multiply $2$ by $- 3$ .

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

Step 2.7

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

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

Step 2.8

Add $3 x^{2} - 6 x$ and $0$ .

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

Step 2.9

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

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

Step 2.10

Simplify with factoring out.

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Step 2.10.1

Multiply $2$ by $- 1$ .

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

Step 2.10.2

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

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Step 2.10.2.1

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

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

Step 2.10.2.2

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

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

Step 2.10.2.3

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

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

Step 3

Cancel the common factors.

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Step 3.1

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

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

Step 3.2

Cancel the common factor.

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

Step 3.3

Rewrite the expression.

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Simplify the numerator.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 4.3.1.2

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

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Step 4.3.1.2.1

Move $x^{2}$ .

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

Step 4.3.1.2.2

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

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Step 4.3.1.2.2.1

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

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

Step 4.3.1.2.2.2

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

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

Step 4.3.1.2.3

Add $2$ and $1$ .

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

Step 4.3.1.3

Rewrite using the commutative property of multiplication.

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

Step 4.3.1.4

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

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Step 4.3.1.4.1

Move $x$ .

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

Step 4.3.1.4.2

Multiply $x$ by $x$ .

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

Step 4.3.1.5

Multiply $- 3$ by $- 2$ .

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

Step 4.3.1.6

Multiply $- 2$ by $4$ .

$\frac{3 x^{3} - 6 x^{2} - 2 x^{3} + 6 x^{2} - 8}{x^{3}}$

Step 4.3.2

Combine the opposite terms in $3 x^{3} - 6 x^{2} - 2 x^{3} + 6 x^{2} - 8$ .

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Step 4.3.2.1

Add $- 6 x^{2}$ and $6 x^{2}$ .

$\frac{3 x^{3} - 2 x^{3} + 0 - 8}{x^{3}}$

Step 4.3.2.2

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

$\frac{3 x^{3} - 2 x^{3} - 8}{x^{3}}$

Step 4.3.3

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

$\frac{x^{3} - 8}{x^{3}}$

Step 4.4

Simplify the numerator.

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Step 4.4.1

Rewrite $8$ as $2^{3}$ .

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

Step 4.4.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 2$ .

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

Step 4.4.3

Simplify.

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Step 4.4.3.1

Move $2$ to the left of $x$ .

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

Step 4.4.3.2

Raise $2$ to the power of $2$ .

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