Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

The derivative of $\arccos (u_{1})$ with respect to $u_{1}$ is $- \frac{1}{\sqrt{1 - {u_{1}}^{2}}}$ .

$- \frac{1}{\sqrt{1 - {u_{1}}^{2}}} \frac{d}{d x} [\frac{1}{1 + x^{2}}]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\frac{1}{1 + x^{2}}$ .

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Differentiate.

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Step 4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2

Combine fractions.

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Step 4.2.1

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

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

Step 4.2.2

Combine ${(1 + x^{2})}^{- 2}$ and $\frac{1}{\sqrt{1 - {(\frac{1}{1 + x^{2}})}^{2}}}$ .

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

Step 4.2.3

Move ${(1 + x^{2})}^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Add $0$ and $\frac{d}{d x} [x^{2}]$ .

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

Step 4.6

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

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

Step 4.7

Combine fractions.

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Step 4.7.1

Combine $2$ and $\frac{1}{\sqrt{1 - {(\frac{1}{1 + x^{2}})}^{2}} {(1 + x^{2})}^{2}}$ .

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

Step 4.7.2

Combine $\frac{2}{\sqrt{1 - {(\frac{1}{1 + x^{2}})}^{2}} {(1 + x^{2})}^{2}}$ and $x$ .

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

Step 5

Simplify.

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Step 5.1

Apply the product rule to $\frac{1}{1 + x^{2}}$ .

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

Step 5.2

One to any power is one.

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

Step 5.3

Reorder terms.

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

Step 5.4

Simplify the denominator.

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Step 5.4.1

Write $1$ as a fraction with a common denominator.

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

Step 5.4.2

Combine the numerators over the common denominator.

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

Step 5.4.3

Reorder terms.

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

Step 5.4.4

Rewrite $\frac{{(1 + x^{2})}^{2} - 1}{{(x^{2} + 1)}^{2}}$ in a factored form.

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Step 5.4.4.1

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

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

Step 5.4.4.2

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

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

Step 5.4.4.3

Simplify.

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Step 5.4.4.3.1

Add $1$ and $1$ .

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

Step 5.4.4.3.2

Subtract $1$ from $1$ .

$\frac{2 x}{{(1 + x^{2})}^{2} \sqrt{\frac{(x^{2} + 2) (x^{2} + 0)}{{(x^{2} + 1)}^{2}}}}$

Step 5.4.4.3.3

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

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

Step 5.4.5

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

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Step 5.4.5.1

Factor the perfect power $x^{2}$ out of $(x^{2} + 2) x^{2}$ .

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

Step 5.4.5.2

Factor the perfect power ${(x^{2} + 1)}^{2}$ out of ${(x^{2} + 1)}^{2}$ .

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

Step 5.4.5.3

Rearrange the fraction $\frac{x^{2} (x^{2} + 2)}{{(x^{2} + 1)}^{2} \cdot 1}$ .

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

Step 5.4.6

Pull terms out from under the radical.

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

Step 5.4.7

Combine $\frac{x}{x^{2} + 1}$ and $\sqrt{x^{2} + 2}$ .

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

Step 5.5

Combine ${(1 + x^{2})}^{2}$ and $\frac{x \sqrt{x^{2} + 2}}{x^{2} + 1}$ .

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

Step 5.6

Simplify the denominator.

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Step 5.6.1

Cancel the common factor of ${(1 + x^{2})}^{2}$ and $x^{2} + 1$ .

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Step 5.6.1.1

Reorder terms.

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

Step 5.6.1.2

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

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

Step 5.6.1.3

Cancel the common factors.

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Step 5.6.1.3.1

Multiply by $1$ .

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

Step 5.6.1.3.2

Cancel the common factor.

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

Step 5.6.1.3.3

Rewrite the expression.

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

Step 5.6.1.3.4

Divide $(x^{2} + 1) x \sqrt{x^{2} + 2}$ by $1$ .

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

Step 5.6.2

Apply the distributive property.

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

Step 5.6.3

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

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Step 5.6.3.1

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

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Step 5.6.3.1.1

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

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

Step 5.6.3.1.2

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

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

Step 5.6.3.2

Add $2$ and $1$ .

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

Step 5.6.4

Multiply $x$ by $1$ .

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

Step 5.6.5

Apply the distributive property.

$\frac{2 x}{x^{3} \sqrt{x^{2} + 2} + x \sqrt{x^{2} + 2}}$

Step 5.6.6

Factor $x \sqrt{x^{2} + 2}$ out of $x^{3} \sqrt{x^{2} + 2} + x \sqrt{x^{2} + 2}$ .

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Step 5.6.6.1

Factor $x \sqrt{x^{2} + 2}$ out of $x^{3} \sqrt{x^{2} + 2}$ .

$\frac{2 x}{x \sqrt{x^{2} + 2} (x^{2}) + x \sqrt{x^{2} + 2}}$

Step 5.6.6.2

Factor $x \sqrt{x^{2} + 2}$ out of $x \sqrt{x^{2} + 2}$ .

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

Step 5.6.6.3

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

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

Step 5.7

Cancel the common factor of $x$ .

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Step 5.7.1

Cancel the common factor.

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

Step 5.7.2

Rewrite the expression.

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

Step 5.8

Multiply $\frac{2}{\sqrt{x^{2} + 2} (x^{2} + 1)}$ by $\frac{\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}}$ .

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

Step 5.9

Combine and simplify the denominator.

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Step 5.9.1

Multiply $\frac{2}{\sqrt{x^{2} + 2} (x^{2} + 1)}$ by $\frac{\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}}$ .

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

Step 5.9.2

Move $\sqrt{x^{2} + 2}$ .

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

Step 5.9.3

Raise $\sqrt{x^{2} + 2}$ to the power of $1$ .

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

Step 5.9.4

Raise $\sqrt{x^{2} + 2}$ to the power of $1$ .

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

Step 5.9.5

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

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

Step 5.9.6

Add $1$ and $1$ .

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

Step 5.9.7

Rewrite ${\sqrt{x^{2} + 2}}^{2}$ as $x^{2} + 2$ .

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Step 5.9.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 2}$ as ${(x^{2} + 2)}^{\frac{1}{2}}$ .

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

Step 5.9.7.2

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

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

Step 5.9.7.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 5.9.7.4

Cancel the common factor of $2$ .

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Step 5.9.7.4.1

Cancel the common factor.

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

Step 5.9.7.4.2

Rewrite the expression.

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

Step 5.9.7.5

Simplify.

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