Calculus Examples

$y = arccsc (x^{4} + 1)$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = arccsc (x)$ and $g (x) = x^{4} + 1$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x^{4} + 1$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $x^{4} + 1$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{4} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [1]$ .

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

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

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

Step 2.3

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

$- \frac{1}{(x^{4} + 1) \sqrt{{(x^{4} + 1)}^{2} - 1}} (4 x^{3} + 0)$

Step 2.4

Combine fractions.

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Step 2.4.1

Add $4 x^{3}$ and $0$ .

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

Step 2.4.2

Multiply $4$ by $- 1$ .

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

Step 2.4.3

Combine $- 4$ and $\frac{1}{(x^{4} + 1) \sqrt{{(x^{4} + 1)}^{2} - 1}}$ .

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

Step 2.4.4

Combine $\frac{- 4}{(x^{4} + 1) \sqrt{{(x^{4} + 1)}^{2} - 1}}$ and $x^{3}$ .

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

Step 2.4.5

Move the negative in front of the fraction.

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

Step 3

Simplify.

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Step 3.1

Simplify the denominator.

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Step 3.1.1

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

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

Step 3.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{4} + 1$ and $b = 1$ .

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

Step 3.1.3

Simplify.

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Step 3.1.3.1

Add $1$ and $1$ .

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

Step 3.1.3.2

Subtract $1$ from $1$ .

$- \frac{4 x^{3}}{(x^{4} + 1) \sqrt{(x^{4} + 2) (x^{4} + 0)}}$

Step 3.1.3.3

Add $x^{4}$ and $0$ .

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

Step 3.1.4

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

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Step 3.1.4.1

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

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

Step 3.1.4.2

Reorder $x^{4} + 2$ and ${(x^{2})}^{2}$ .

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

Step 3.1.4.3

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

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

Step 3.1.5

Pull terms out from under the radical.

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

Step 3.1.6

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

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Step 3.1.6.1

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

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

Step 3.1.6.2

Multiply $2$ by $2$ .

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

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

Step 3.2

Cancel the common factor of $x^{3}$ and $x^{2}$ .

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Step 3.2.1

Factor $x^{2}$ out of $4 x^{3}$ .

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

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

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

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

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

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

Step 3.3

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

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

Step 3.4

Combine and simplify the denominator.

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Step 3.4.1

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

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

Step 3.4.2

Move $\sqrt{x^{4} + 2}$ .

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

Step 3.4.3

Raise $\sqrt{x^{4} + 2}$ to the power of $1$ .

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

Step 3.4.4

Raise $\sqrt{x^{4} + 2}$ to the power of $1$ .

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

Step 3.4.5

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

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

Step 3.4.6

Add $1$ and $1$ .

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

Step 3.4.7

Rewrite ${\sqrt{x^{4} + 2}}^{2}$ as $x^{4} + 2$ .

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Step 3.4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{4} + 2}$ as ${(x^{4} + 2)}^{\frac{1}{2}}$ .

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

Step 3.4.7.2

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

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

Step 3.4.7.3

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

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

Step 3.4.7.4

Cancel the common factor of $2$ .

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Step 3.4.7.4.1

Cancel the common factor.

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

Step 3.4.7.4.2

Rewrite the expression.

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

Step 3.4.7.5

Simplify.

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