Calculus Examples

$y = 2 \csc^{3} (\sqrt{x})$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

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

$\frac{d}{d x} [2 \csc^{3} (x^{\frac{1}{2}})]$

Step 1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 \csc^{3} (x^{\frac{1}{2}})$ with respect to $x$ is $2 \frac{d}{d x} [\csc^{3} (x^{\frac{1}{2}})]$ .

$2 \frac{d}{d x} [\csc^{3} (x^{\frac{1}{2}})]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \csc (x^{\frac{1}{2}})$ .

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Step 2.1

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

$2 (\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d x} [\csc (x^{\frac{1}{2}})])$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 3$ .

$2 (3 {u_{1}}^{2} \frac{d}{d x} [\csc (x^{\frac{1}{2}})])$

Step 2.3

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

$2 (3 \csc^{2} (x^{\frac{1}{2}}) \frac{d}{d x} [\csc (x^{\frac{1}{2}})])$

Step 3

Multiply $3$ by $2$ .

$6 (\csc^{2} (x^{\frac{1}{2}}) \frac{d}{d x} [\csc (x^{\frac{1}{2}})])$

Step 4

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

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Step 4.1

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

$6 \csc^{2} (x^{\frac{1}{2}}) (\frac{d}{d u_{2}} [\csc (u_{2})] \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 4.2

The derivative of $\csc (u_{2})$ with respect to $u_{2}$ is $- \csc (u_{2}) \cot (u_{2})$ .

$6 \csc^{2} (x^{\frac{1}{2}}) (- \csc (u_{2}) \cot (u_{2}) \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 4.3

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

$6 \csc^{2} (x^{\frac{1}{2}}) (- \csc (x^{\frac{1}{2}}) \cot (x^{\frac{1}{2}}) \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 5

Multiply $- 1$ by $6$ .

$- 6 \csc^{2} (x^{\frac{1}{2}}) (\csc (x^{\frac{1}{2}}) \cot (x^{\frac{1}{2}}) \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 6

Raise $\csc (x^{\frac{1}{2}})$ to the power of $1$ .

$- 6 (\csc^{1} (x^{\frac{1}{2}}) \csc^{2} (x^{\frac{1}{2}})) (\cot (x^{\frac{1}{2}}) \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 7

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

$- 6 {\csc (x^{\frac{1}{2}})}^{1 + 2} (\cot (x^{\frac{1}{2}}) \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 8

Add $1$ and $2$ .

$- 6 \csc^{3} (x^{\frac{1}{2}}) (\cot (x^{\frac{1}{2}}) \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 9

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

$- 6 \csc^{3} (x^{\frac{1}{2}}) \cot (x^{\frac{1}{2}}) (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 10

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 6 \csc^{3} (x^{\frac{1}{2}}) \cot (x^{\frac{1}{2}}) (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 11

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

$- 6 \csc^{3} (x^{\frac{1}{2}}) \cot (x^{\frac{1}{2}}) (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 12

Combine the numerators over the common denominator.

$- 6 \csc^{3} (x^{\frac{1}{2}}) \cot (x^{\frac{1}{2}}) (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 13

Simplify the numerator.

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Step 13.1

Multiply $- 1$ by $2$ .

$- 6 \csc^{3} (x^{\frac{1}{2}}) \cot (x^{\frac{1}{2}}) (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 13.2

Subtract $2$ from $1$ .

$- 6 \csc^{3} (x^{\frac{1}{2}}) \cot (x^{\frac{1}{2}}) (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 14

Move the negative in front of the fraction.

$- 6 \csc^{3} (x^{\frac{1}{2}}) \cot (x^{\frac{1}{2}}) (\frac{1}{2} x^{- \frac{1}{2}})$

Step 15

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

$- 6 \csc^{3} (x^{\frac{1}{2}}) \cot (x^{\frac{1}{2}}) \frac{x^{- \frac{1}{2}}}{2}$

Step 16

Combine $\frac{x^{- \frac{1}{2}}}{2}$ and $- 6$ .

$\frac{x^{- \frac{1}{2}} \cdot - 6}{2} \csc^{3} (x^{\frac{1}{2}}) \cot (x^{\frac{1}{2}})$

Step 17

Combine $\frac{x^{- \frac{1}{2}} \cdot - 6}{2}$ and $\csc^{3} (x^{\frac{1}{2}})$ .

$\frac{x^{- \frac{1}{2}} \cdot - 6 \csc^{3} (x^{\frac{1}{2}})}{2} \cot (x^{\frac{1}{2}})$

Step 18

Combine $\frac{x^{- \frac{1}{2}} \cdot - 6 \csc^{3} (x^{\frac{1}{2}})}{2}$ and $\cot (x^{\frac{1}{2}})$ .

$\frac{x^{- \frac{1}{2}} \cdot - 6 \csc^{3} (x^{\frac{1}{2}}) \cot (x^{\frac{1}{2}})}{2}$

Step 19

Simplify the expression.

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Step 19.1

Move $- 6$ to the left of $x^{- \frac{1}{2}}$ .

$\frac{- 6 \cdot x^{- \frac{1}{2}} \csc^{3} (x^{\frac{1}{2}}) \cot (x^{\frac{1}{2}})}{2}$

Step 19.2

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

$\frac{- 6 \csc^{3} (x^{\frac{1}{2}}) \cot (x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}$

Step 20

Factor $2$ out of $- 6 \csc^{3} (x^{\frac{1}{2}}) \cot (x^{\frac{1}{2}})$ .

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

Step 21

Cancel the common factors.

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Step 21.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

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

Step 21.2

Cancel the common factor.

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

Step 21.3

Rewrite the expression.

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

Step 22

Move the negative in front of the fraction.

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