Calculus Examples

$f (x) = 4 arccsc (10 x^{- 1})$

Step 1

Since $4$ is constant with respect to $x$ , the derivative of $4 arccsc (10 x^{- 1})$ with respect to $x$ is $4 \frac{d}{d x} [arccsc (10 x^{- 1})]$ .

$4 \frac{d}{d x} [arccsc (10 x^{- 1})]$

Step 2

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 $g (x) = 10 x^{- 1}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $10 x^{- 1}$ .

$4 (\frac{d}{d u} [arccsc (u)] \frac{d}{d x} [10 x^{- 1}])$

Step 2.2

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

$4 (- \frac{1}{u \sqrt{u^{2} - 1}} \frac{d}{d x} [10 x^{- 1}])$

Step 2.3

Replace all occurrences of $u$ with $10 x^{- 1}$ .

$4 (- \frac{1}{10 x^{- 1} \sqrt{{(10 x^{- 1})}^{2} - 1}} \frac{d}{d x} [10 x^{- 1}])$

Step 3

Differentiate.

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Step 3.1

Factor $10$ out of $10 x^{- 1}$ .

$4 (- \frac{1}{10 x^{- 1} \sqrt{{(10 (x^{- 1}))}^{2} - 1}} \frac{d}{d x} [10 x^{- 1}])$

Step 3.2

Simplify terms.

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Step 3.2.1

Simplify the expression.

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Step 3.2.1.1

Apply the product rule to $10 (x^{- 1})$ .

$4 (- \frac{1}{10 x^{- 1} \sqrt{10^{2} {(x^{- 1})}^{2} - 1}} \frac{d}{d x} [10 x^{- 1}])$

Step 3.2.1.2

Raise $10$ to the power of $2$ .

$4 (- \frac{1}{10 x^{- 1} \sqrt{100 {(x^{- 1})}^{2} - 1}} \frac{d}{d x} [10 x^{- 1}])$

Step 3.2.1.3

Multiply the exponents in ${(x^{- 1})}^{2}$ .

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Step 3.2.1.3.1

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

$4 (- \frac{1}{10 x^{- 1} \sqrt{100 x^{- 1 \cdot 2} - 1}} \frac{d}{d x} [10 x^{- 1}])$

Step 3.2.1.3.2

Multiply $- 1$ by $2$ .

$4 (- \frac{1}{10 x^{- 1} \sqrt{100 x^{- 2} - 1}} \frac{d}{d x} [10 x^{- 1}])$

Step 3.2.1.4

Move $x^{- 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$4 (- \frac{x}{10 \sqrt{100 x^{- 2} - 1}} \frac{d}{d x} [10 x^{- 1}])$

Step 3.2.1.5

Multiply $- 1$ by $4$ .

$- 4 (\frac{x}{10 \sqrt{100 x^{- 2} - 1}} \frac{d}{d x} [10 x^{- 1}])$

Step 3.2.2

Combine $\frac{x}{10 \sqrt{100 x^{- 2} - 1}}$ and $- 4$ .

$\frac{x \cdot - 4}{10 \sqrt{100 x^{- 2} - 1}} \frac{d}{d x} [10 x^{- 1}]$

Step 3.2.3

Move $- 4$ to the left of $x$ .

$\frac{- 4 \cdot x}{10 \sqrt{100 x^{- 2} - 1}} \frac{d}{d x} [10 x^{- 1}]$

Step 3.2.4

Cancel the common factor of $- 4$ and $10$ .

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Step 3.2.4.1

Factor $2$ out of $- 4 x$ .

$\frac{2 (- 2 x)}{10 \sqrt{100 x^{- 2} - 1}} \frac{d}{d x} [10 x^{- 1}]$

Step 3.2.4.2

Cancel the common factors.

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Step 3.2.4.2.1

Factor $2$ out of $10 \sqrt{100 x^{- 2} - 1}$ .

$\frac{2 (- 2 x)}{2 (5 \sqrt{100 x^{- 2} - 1})} \frac{d}{d x} [10 x^{- 1}]$

Step 3.2.4.2.2

Cancel the common factor.

$\frac{2 (- 2 x)}{2 (5 \sqrt{100 x^{- 2} - 1})} \frac{d}{d x} [10 x^{- 1}]$

Step 3.2.4.2.3

Rewrite the expression.

$\frac{- 2 x}{5 \sqrt{100 x^{- 2} - 1}} \frac{d}{d x} [10 x^{- 1}]$

Step 3.2.5

Move the negative in front of the fraction.

$- \frac{2 x}{5 \sqrt{100 x^{- 2} - 1}} \frac{d}{d x} [10 x^{- 1}]$

Step 3.3

Since $10$ is constant with respect to $x$ , the derivative of $10 x^{- 1}$ with respect to $x$ is $10 \frac{d}{d x} [x^{- 1}]$ .

$- \frac{2 x}{5 \sqrt{100 x^{- 2} - 1}} (10 \frac{d}{d x} [x^{- 1}])$

Step 3.4

Simplify terms.

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Step 3.4.1

Multiply $10$ by $- 1$ .

$- 10 \frac{2 x}{5 \sqrt{100 x^{- 2} - 1}} \frac{d}{d x} [x^{- 1}]$

Step 3.4.2

Combine $- 10$ and $\frac{2 x}{5 \sqrt{100 x^{- 2} - 1}}$ .

$\frac{- 10 (2 x)}{5 \sqrt{100 x^{- 2} - 1}} \frac{d}{d x} [x^{- 1}]$

Step 3.4.3

Multiply $2$ by $- 10$ .

$\frac{- 20 x}{5 \sqrt{100 x^{- 2} - 1}} \frac{d}{d x} [x^{- 1}]$

Step 3.4.4

Cancel the common factor of $- 20$ and $5$ .

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Step 3.4.4.1

Factor $5$ out of $- 20 x$ .

$\frac{5 (- 4 x)}{5 \sqrt{100 x^{- 2} - 1}} \frac{d}{d x} [x^{- 1}]$

Step 3.4.4.2

Cancel the common factors.

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Step 3.4.4.2.1

Factor $5$ out of $5 \sqrt{100 x^{- 2} - 1}$ .

$\frac{5 (- 4 x)}{5 (\sqrt{100 x^{- 2} - 1})} \frac{d}{d x} [x^{- 1}]$

Step 3.4.4.2.2

Cancel the common factor.

$\frac{5 (- 4 x)}{5 \sqrt{100 x^{- 2} - 1}} \frac{d}{d x} [x^{- 1}]$

Step 3.4.4.2.3

Rewrite the expression.

$\frac{- 4 x}{\sqrt{100 x^{- 2} - 1}} \frac{d}{d x} [x^{- 1}]$

Step 3.4.5

Move the negative in front of the fraction.

$- \frac{4 x}{\sqrt{100 x^{- 2} - 1}} \frac{d}{d x} [x^{- 1}]$

Step 3.5

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

$- \frac{4 x}{\sqrt{100 x^{- 2} - 1}} (- x^{- 2})$

Step 3.6

Combine fractions.

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Step 3.6.1

Multiply $- 1$ by $- 1$ .

$1 \frac{4 x}{\sqrt{100 x^{- 2} - 1}} x^{- 2}$

Step 3.6.2

Multiply $\frac{4 x}{\sqrt{100 x^{- 2} - 1}}$ by $1$ .

$\frac{4 x}{\sqrt{100 x^{- 2} - 1}} x^{- 2}$

Step 3.6.3

Combine $\frac{4 x}{\sqrt{100 x^{- 2} - 1}}$ and $x^{- 2}$ .

$\frac{4 x \cdot x^{- 2}}{\sqrt{100 x^{- 2} - 1}}$

Step 4

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

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Step 4.1

Move $x^{- 2}$ .

$\frac{4 (x^{- 2} x)}{\sqrt{100 x^{- 2} - 1}}$

Step 4.2

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

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Step 4.2.1

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

$\frac{4 (x^{- 2} x^{1})}{\sqrt{100 x^{- 2} - 1}}$

Step 4.2.2

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

$\frac{4 x^{- 2 + 1}}{\sqrt{100 x^{- 2} - 1}}$

Step 4.3

Add $- 2$ and $1$ .

$\frac{4 x^{- 1}}{\sqrt{100 x^{- 2} - 1}}$

Step 5

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

$\frac{4}{\sqrt{100 x^{- 2} - 1} x}$

Step 6

Reorder terms.

$\frac{4}{x \sqrt{100 x^{- 2} - 1}}$