Calculus Examples

$y = \frac{{(x^{2} + 1)}^{2}}{2 x^{2}}$

Step 1

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

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

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Multiply $2$ by $2$ .

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

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{2} + 1$ .

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Step 4.1

To apply the Chain Rule, set $u$ as $x^{2} + 1$ .

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

Step 4.2

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

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

Step 4.3

Replace all occurrences of $u$ with $x^{2} + 1$ .

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

Step 5

Differentiate.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Simplify the expression.

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Step 5.4.1

Add $2 x$ and $0$ .

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

Step 5.4.2

Multiply $2$ by $2$ .

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

Step 6

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

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Step 6.1

Move $x$ .

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.3

Add $1$ and $2$ .

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

Step 7

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

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

Step 8

Combine fractions.

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Step 8.1

Multiply $2$ by $- 1$ .

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

Step 8.2

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

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

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Apply the distributive property.

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

Step 9.3

Simplify the numerator.

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Step 9.3.1

Simplify each term.

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Step 9.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 9.3.1.2

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 9.3.1.2.1

Move $x^{2}$ .

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

Step 9.3.1.2.2

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

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

Step 9.3.1.2.3

Add $2$ and $3$ .

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

Step 9.3.1.3

Multiply $4$ by $1$ .

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

Step 9.3.1.4

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

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

Step 9.3.1.5

Rewrite ${(x^{2} + 1)}^{2}$ as $(x^{2} + 1) (x^{2} + 1)$ .

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

Step 9.3.1.6

Expand $(x^{2} + 1) (x^{2} + 1)$ using the FOIL Method.

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Step 9.3.1.6.1

Apply the distributive property.

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

Step 9.3.1.6.2

Apply the distributive property.

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

Step 9.3.1.6.3

Apply the distributive property.

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

Step 9.3.1.7

Simplify and combine like terms.

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Step 9.3.1.7.1

Simplify each term.

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Step 9.3.1.7.1.1

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

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Step 9.3.1.7.1.1.1

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

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

Step 9.3.1.7.1.1.2

Add $2$ and $2$ .

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

Step 9.3.1.7.1.2

Multiply $x^{2}$ by $1$ .

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

Step 9.3.1.7.1.3

Multiply $x^{2}$ by $1$ .

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

Step 9.3.1.7.1.4

Multiply $1$ by $1$ .

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

Step 9.3.1.7.2

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

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

Step 9.3.1.8

Apply the distributive property.

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

Step 9.3.1.9

Simplify.

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Step 9.3.1.9.1

Multiply $2$ by $- 2$ .

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

Step 9.3.1.9.2

Multiply $- 2$ by $1$ .

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

Step 9.3.1.10

Apply the distributive property.

$\frac{4 x^{5} + 4 x^{3} - 2 x^{4} x - 4 x^{2} x - 2 x}{2 x^{4}}$

Step 9.3.1.11

Simplify.

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Step 9.3.1.11.1

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 9.3.1.11.1.1

Move $x$ .

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

Step 9.3.1.11.1.2

Multiply $x$ by $x^{4}$ .

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Step 9.3.1.11.1.2.1

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

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

Step 9.3.1.11.1.2.2

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

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

Step 9.3.1.11.1.3

Add $1$ and $4$ .

$\frac{4 x^{5} + 4 x^{3} - 2 x^{5} - 4 x^{2} x - 2 x}{2 x^{4}}$

Step 9.3.1.11.2

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

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Step 9.3.1.11.2.1

Move $x$ .

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

Step 9.3.1.11.2.2

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

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Step 9.3.1.11.2.2.1

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

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

Step 9.3.1.11.2.2.2

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

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

Step 9.3.1.11.2.3

Add $1$ and $2$ .

$\frac{4 x^{5} + 4 x^{3} - 2 x^{5} - 4 x^{3} - 2 x}{2 x^{4}}$

Step 9.3.2

Combine the opposite terms in $4 x^{5} + 4 x^{3} - 2 x^{5} - 4 x^{3} - 2 x$ .

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Step 9.3.2.1

Subtract $4 x^{3}$ from $4 x^{3}$ .

$\frac{4 x^{5} - 2 x^{5} + 0 - 2 x}{2 x^{4}}$

Step 9.3.2.2

Add $4 x^{5} - 2 x^{5}$ and $0$ .

$\frac{4 x^{5} - 2 x^{5} - 2 x}{2 x^{4}}$

Step 9.3.3

Subtract $2 x^{5}$ from $4 x^{5}$ .

$\frac{2 x^{5} - 2 x}{2 x^{4}}$

Step 9.4

Simplify the numerator.

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Step 9.4.1

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

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Step 9.4.1.1

Factor $2 x$ out of $2 x^{5}$ .

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

Step 9.4.1.2

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

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

Step 9.4.1.3

Factor $2 x$ out of $2 x (x^{4}) + 2 x (- 1)$ .

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

Step 9.4.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

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

Step 9.4.3

Rewrite $1$ as $1^{2}$ .

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

Step 9.4.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 1$ .

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

Step 9.4.5

Simplify.

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Step 9.4.5.1

Rewrite $1$ as $1^{2}$ .

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

Step 9.4.5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

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

Step 9.5

Cancel the common factor of $2$ .

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Step 9.5.1

Cancel the common factor.

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

Step 9.5.2

Rewrite the expression.

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

Step 9.6

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

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Step 9.6.1

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

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

Step 9.6.2

Cancel the common factors.

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Step 9.6.2.1

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

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

Step 9.6.2.2

Cancel the common factor.

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

Step 9.6.2.3

Rewrite the expression.

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

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