Calculus Examples

$f (x) = - 6 \csc (\sqrt{5 x + 3})$

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 $\sqrt{5 x + 3}$ as ${(5 x + 3)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [- 6 \csc ({(5 x + 3)}^{\frac{1}{2}})]$

Step 1.2

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 \csc ({(5 x + 3)}^{\frac{1}{2}})$ with respect to $x$ is $- 6 \frac{d}{d x} [\csc ({(5 x + 3)}^{\frac{1}{2}})]$ .

$- 6 \frac{d}{d x} [\csc ({(5 x + 3)}^{\frac{1}{2}})]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \csc (x)$ and $g (x) = {(5 x + 3)}^{\frac{1}{2}}$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as ${(5 x + 3)}^{\frac{1}{2}}$ .

$- 6 (\frac{d}{d u_{1}} [\csc (u_{1})] \frac{d}{d x} [{(5 x + 3)}^{\frac{1}{2}}])$

Step 2.2

The derivative of $\csc (u_{1})$ with respect to $u_{1}$ is $- \csc (u_{1}) \cot (u_{1})$ .

$- 6 (- \csc (u_{1}) \cot (u_{1}) \frac{d}{d x} [{(5 x + 3)}^{\frac{1}{2}}])$

Step 2.3

Replace all occurrences of $u_{1}$ with ${(5 x + 3)}^{\frac{1}{2}}$ .

$- 6 (- \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}}) \frac{d}{d x} [{(5 x + 3)}^{\frac{1}{2}}])$

Step 3

Multiply $- 1$ by $- 6$ .

$6 (\csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}}) \frac{d}{d x} [{(5 x + 3)}^{\frac{1}{2}}])$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 5 x + 3$ .

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Step 4.1

To apply the Chain Rule, set $u_{2}$ as $5 x + 3$ .

$6 \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}}) (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{2}}] \frac{d}{d x} [5 x + 3])$

Step 4.2

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

$6 \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}}) (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d x} [5 x + 3])$

Step 4.3

Replace all occurrences of $u_{2}$ with $5 x + 3$ .

$6 \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}}) (\frac{1}{2} {(5 x + 3)}^{\frac{1}{2} - 1} \frac{d}{d x} [5 x + 3])$

Step 5

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

$6 \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}}) (\frac{1}{2} {(5 x + 3)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [5 x + 3])$

Step 6

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

$6 \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}}) (\frac{1}{2} {(5 x + 3)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [5 x + 3])$

Step 7

Combine the numerators over the common denominator.

$6 \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}}) (\frac{1}{2} {(5 x + 3)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [5 x + 3])$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $2$ .

$6 \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}}) (\frac{1}{2} {(5 x + 3)}^{\frac{1 - 2}{2}} \frac{d}{d x} [5 x + 3])$

Step 8.2

Subtract $2$ from $1$ .

$6 \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}}) (\frac{1}{2} {(5 x + 3)}^{\frac{- 1}{2}} \frac{d}{d x} [5 x + 3])$

Step 9

Move the negative in front of the fraction.

$6 \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}}) (\frac{1}{2} {(5 x + 3)}^{- \frac{1}{2}} \frac{d}{d x} [5 x + 3])$

Step 10

Combine $\frac{1}{2}$ and ${(5 x + 3)}^{- \frac{1}{2}}$ .

$6 \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}}) (\frac{{(5 x + 3)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [5 x + 3])$

Step 11

Move ${(5 x + 3)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$6 \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}}) (\frac{1}{2 {(5 x + 3)}^{\frac{1}{2}}} \frac{d}{d x} [5 x + 3])$

Step 12

Combine $\frac{1}{2 {(5 x + 3)}^{\frac{1}{2}}}$ and $6$ .

$\frac{6}{2 {(5 x + 3)}^{\frac{1}{2}}} \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}}) \frac{d}{d x} [5 x + 3]$

Step 13

Combine $\frac{6}{2 {(5 x + 3)}^{\frac{1}{2}}}$ and $\csc ({(5 x + 3)}^{\frac{1}{2}})$ .

$\frac{6 \csc ({(5 x + 3)}^{\frac{1}{2}})}{2 {(5 x + 3)}^{\frac{1}{2}}} \cot ({(5 x + 3)}^{\frac{1}{2}}) \frac{d}{d x} [5 x + 3]$

Step 14

Combine $\frac{6 \csc ({(5 x + 3)}^{\frac{1}{2}})}{2 {(5 x + 3)}^{\frac{1}{2}}}$ and $\cot ({(5 x + 3)}^{\frac{1}{2}})$ .

$\frac{6 \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}})}{2 {(5 x + 3)}^{\frac{1}{2}}} \frac{d}{d x} [5 x + 3]$

Step 15

Factor $2$ out of $6 \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}})$ .

$\frac{2 (3 \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}}))}{2 {(5 x + 3)}^{\frac{1}{2}}} \frac{d}{d x} [5 x + 3]$

Step 16

Cancel the common factors.

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Step 16.1

Factor $2$ out of $2 {(5 x + 3)}^{\frac{1}{2}}$ .

$\frac{2 (3 \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}}))}{2 ({(5 x + 3)}^{\frac{1}{2}})} \frac{d}{d x} [5 x + 3]$

Step 16.2

Cancel the common factor.

$\frac{2 (3 \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}}))}{2 {(5 x + 3)}^{\frac{1}{2}}} \frac{d}{d x} [5 x + 3]$

Step 16.3

Rewrite the expression.

$\frac{3 \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}})}{{(5 x + 3)}^{\frac{1}{2}}} \frac{d}{d x} [5 x + 3]$

Step 17

By the Sum Rule, the derivative of $5 x + 3$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [3]$ .

$\frac{3 \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}})}{{(5 x + 3)}^{\frac{1}{2}}} (\frac{d}{d x} [5 x] + \frac{d}{d x} [3])$

Step 18

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$\frac{3 \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}})}{{(5 x + 3)}^{\frac{1}{2}}} (5 \frac{d}{d x} [x] + \frac{d}{d x} [3])$

Step 19

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

$\frac{3 \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}})}{{(5 x + 3)}^{\frac{1}{2}}} (5 \cdot 1 + \frac{d}{d x} [3])$

Step 20

Multiply $5$ by $1$ .

$\frac{3 \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}})}{{(5 x + 3)}^{\frac{1}{2}}} (5 + \frac{d}{d x} [3])$

Step 21

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{3 \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}})}{{(5 x + 3)}^{\frac{1}{2}}} (5 + 0)$

Step 22

Combine fractions.

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Step 22.1

Add $5$ and $0$ .

$\frac{3 \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}})}{{(5 x + 3)}^{\frac{1}{2}}} \cdot 5$

Step 22.2

Combine $\frac{3 \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}})}{{(5 x + 3)}^{\frac{1}{2}}}$ and $5$ .

$\frac{3 \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}}) \cdot 5}{{(5 x + 3)}^{\frac{1}{2}}}$

Step 22.3

Simplify the expression.

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Step 22.3.1

Multiply $5$ by $3$ .

$\frac{15 \csc ({(5 x + 3)}^{\frac{1}{2}}) \cot ({(5 x + 3)}^{\frac{1}{2}})}{{(5 x + 3)}^{\frac{1}{2}}}$

Step 22.3.2

Reorder terms.

$\frac{15 \cot ({(5 x + 3)}^{\frac{1}{2}}) \csc ({(5 x + 3)}^{\frac{1}{2}})}{{(5 x + 3)}^{\frac{1}{2}}}$