Calculus Examples

${(\csc (7 x - 1))}^{\frac{2}{3}}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{2}{3}}$ and $g (x) = \csc (7 x - 1)$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\csc (7 x - 1)$ .

$\frac{d}{d u_{1}} [{u_{1}}^{\frac{2}{3}}] \frac{d}{d x} [\csc (7 x - 1)]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = \frac{2}{3}$ .

$\frac{2}{3} {u_{1}}^{\frac{2}{3} - 1} \frac{d}{d x} [\csc (7 x - 1)]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\csc (7 x - 1)$ .

$\frac{2}{3} {\csc (7 x - 1)}^{\frac{2}{3} - 1} \frac{d}{d x} [\csc (7 x - 1)]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \csc (x)$ and $g (x) = 7 x - 1$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $7 x - 1$ .

$\frac{2}{3} {\csc (7 x - 1)}^{\frac{2}{3} - 1} (\frac{d}{d u_{2}} [\csc (u_{2})] \frac{d}{d x} [7 x - 1])$

Step 2.2

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

$\frac{2}{3} {\csc (7 x - 1)}^{\frac{2}{3} - 1} (- \csc (u_{2}) \cot (u_{2}) \frac{d}{d x} [7 x - 1])$

Step 2.3

Replace all occurrences of $u_{2}$ with $7 x - 1$ .

$\frac{2}{3} {\csc (7 x - 1)}^{\frac{2}{3} - 1} (- \csc (7 x - 1) \cot (7 x - 1) \frac{d}{d x} [7 x - 1])$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $7 x - 1$ with respect to $x$ is $\frac{d}{d x} [7 x] + \frac{d}{d x} [- 1]$ .

$\frac{2}{3} {\csc (7 x - 1)}^{\frac{2}{3} - 1} (- \csc (7 x - 1) \cot (7 x - 1) (\frac{d}{d x} [7 x] + \frac{d}{d x} [- 1]))$

Step 3.2

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$\frac{2}{3} {\csc (7 x - 1)}^{\frac{2}{3} - 1} (- \csc (7 x - 1) \cot (7 x - 1) (7 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]))$

Step 3.3

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

$\frac{2}{3} {\csc (7 x - 1)}^{\frac{2}{3} - 1} (- \csc (7 x - 1) \cot (7 x - 1) (7 \cdot 1 + \frac{d}{d x} [- 1]))$

Step 3.4

Multiply $7$ by $1$ .

$\frac{2}{3} {\csc (7 x - 1)}^{\frac{2}{3} - 1} (- \csc (7 x - 1) \cot (7 x - 1) (7 + \frac{d}{d x} [- 1]))$

Step 3.5

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

$\frac{2}{3} {\csc (7 x - 1)}^{\frac{2}{3} - 1} (- \csc (7 x - 1) \cot (7 x - 1) (7 + 0))$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $7$ and $0$ .

$\frac{2}{3} {\csc (7 x - 1)}^{\frac{2}{3} - 1} (- \csc (7 x - 1) \cot (7 x - 1) \cdot 7)$

Step 3.6.2

Multiply $7$ by $- 1$ .

$\frac{2}{3} {\csc (7 x - 1)}^{\frac{2}{3} - 1} (- 7 \csc (7 x - 1) \cot (7 x - 1))$

Step 3.6.3

Reorder the factors of $- 7 \csc (7 x - 1) \cot (7 x - 1)$ .

$\frac{2}{3} {\csc (7 x - 1)}^{\frac{2}{3} - 1} (- 7 \cot (7 x - 1) \csc (7 x - 1))$

Step 4

Simplify.

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Step 4.1

Combine terms.

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Step 4.1.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2}{3} {\csc (7 x - 1)}^{\frac{2}{3} - 1 \cdot \frac{3}{3}} (- 7 \cot (7 x - 1) \csc (7 x - 1))$

Step 4.1.2

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{2}{3} {\csc (7 x - 1)}^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}} (- 7 \cot (7 x - 1) \csc (7 x - 1))$

Step 4.1.3

Combine the numerators over the common denominator.

$\frac{2}{3} {\csc (7 x - 1)}^{\frac{2 - 1 \cdot 3}{3}} (- 7 \cot (7 x - 1) \csc (7 x - 1))$

Step 4.1.4

Simplify the numerator.

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Step 4.1.4.1

Multiply $- 1$ by $3$ .

$\frac{2}{3} {\csc (7 x - 1)}^{\frac{2 - 3}{3}} (- 7 \cot (7 x - 1) \csc (7 x - 1))$

Step 4.1.4.2

Subtract $3$ from $2$ .

$\frac{2}{3} {\csc (7 x - 1)}^{\frac{- 1}{3}} (- 7 \cot (7 x - 1) \csc (7 x - 1))$

Step 4.1.5

Move the negative in front of the fraction.

$\frac{2}{3} {\csc (7 x - 1)}^{- \frac{1}{3}} (- 7 \cot (7 x - 1) \csc (7 x - 1))$

Step 4.1.6

Combine $\frac{2}{3}$ and ${\csc (7 x - 1)}^{- \frac{1}{3}}$ .

$\frac{2 {\csc (7 x - 1)}^{- \frac{1}{3}}}{3} (- 7 \cot (7 x - 1) \csc (7 x - 1))$

Step 4.1.7

Combine $- 7$ and $\frac{2 {\csc (7 x - 1)}^{- \frac{1}{3}}}{3}$ .

$\frac{- 7 (2 {\csc (7 x - 1)}^{- \frac{1}{3}})}{3} (\cot (7 x - 1) \csc (7 x - 1))$

Step 4.1.8

Multiply $2$ by $- 7$ .

$\frac{- 14 {\csc (7 x - 1)}^{- \frac{1}{3}}}{3} (\cot (7 x - 1) \csc (7 x - 1))$

Step 4.1.9

Combine $\cot (7 x - 1)$ and $\frac{- 14 {\csc (7 x - 1)}^{- \frac{1}{3}}}{3}$ .

$\frac{\cot (7 x - 1) (- 14 {\csc (7 x - 1)}^{- \frac{1}{3}})}{3} \csc (7 x - 1)$

Step 4.1.10

Combine $\frac{\cot (7 x - 1) (- 14 {\csc (7 x - 1)}^{- \frac{1}{3}})}{3}$ and $\csc (7 x - 1)$ .

$\frac{\cot (7 x - 1) (- 14 {\csc (7 x - 1)}^{- \frac{1}{3}}) \csc (7 x - 1)}{3}$

Step 4.1.11

Raise $\csc (7 x - 1)$ to the power of $1$ .

$\frac{\cot (7 x - 1) (- 14 (\csc^{1} (7 x - 1) {\csc (7 x - 1)}^{- \frac{1}{3}}))}{3}$

Step 4.1.12

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

$\frac{\cot (7 x - 1) (- 14 {\csc (7 x - 1)}^{1 - \frac{1}{3}})}{3}$

Step 4.1.13

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

$\frac{\cot (7 x - 1) (- 14 {\csc (7 x - 1)}^{\frac{3}{3} - \frac{1}{3}})}{3}$

Step 4.1.14

Combine the numerators over the common denominator.

$\frac{\cot (7 x - 1) (- 14 {\csc (7 x - 1)}^{\frac{3 - 1}{3}})}{3}$

Step 4.1.15

Subtract $1$ from $3$ .

$\frac{\cot (7 x - 1) (- 14 {\csc (7 x - 1)}^{\frac{2}{3}})}{3}$

Step 4.1.16

Move the negative in front of the fraction.

$- \frac{\cot (7 x - 1) \cdot 14 {\csc (7 x - 1)}^{\frac{2}{3}}}{3}$

Step 4.1.17

Move $14$ to the left of $\cot (7 x - 1)$ .

$- \frac{14 \cot (7 x - 1) {\csc (7 x - 1)}^{\frac{2}{3}}}{3}$

Step 4.2

Reorder factors in $- \frac{14 \cot (7 x - 1) {\csc (7 x - 1)}^{\frac{2}{3}}}{3}$ .

$- \frac{14 {\csc (7 x - 1)}^{\frac{2}{3}} \cot (7 x - 1)}{3}$