Calculus Examples

$\frac{x^{2} + 1}{x^{2}}$

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

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

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

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

Step 2.1.2

Multiply $2$ by $2$ .

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

Step 2.2

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Add $2 x$ and $0$ .

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

Step 3

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

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Step 3.1

Move $x$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.3

Add $1$ and $2$ .

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

Step 4

Move $2$ to the left of $x^{3}$ .

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

Step 5

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

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

Step 6

Multiply $2$ by $- 1$ .

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

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Simplify the numerator.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

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

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Step 7.3.1.1.1

Move $x$ .

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

Step 7.3.1.1.2

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

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Step 7.3.1.1.2.1

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

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

Step 7.3.1.1.2.2

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

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

Step 7.3.1.1.3

Add $1$ and $2$ .

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

Step 7.3.1.2

Multiply $- 2$ by $1$ .

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

Step 7.3.2

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

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

Step 7.3.3

Subtract $2 x$ from $0$ .

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

Step 7.4

Combine terms.

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Step 7.4.1

Cancel the common factor of $x$ and $x^{4}$ .

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Step 7.4.1.1

Factor $x$ out of $- 2 x$ .

$\frac{x \cdot - 2}{x^{4}}$

Step 7.4.1.2

Cancel the common factors.

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Step 7.4.1.2.1

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

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

Step 7.4.1.2.2

Cancel the common factor.

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

Step 7.4.1.2.3

Rewrite the expression.

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

Step 7.4.2

Move the negative in front of the fraction.

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