Calculus Examples

$\ln (\csc (5 x))$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\csc (5 x)$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [\csc (5 x)]$

Step 1.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

$\frac{1}{u_{1}} \frac{d}{d x} [\csc (5 x)]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\csc (5 x)$ .

$\frac{1}{\csc (5 x)} \frac{d}{d x} [\csc (5 x)]$

Step 2

Rewrite $\csc (5 x)$ in terms of sines and cosines.

$\frac{1}{\frac{1}{\sin (5 x)}} \frac{d}{d x} [\csc (5 x)]$

Step 3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (5 x)}$ .

$1 \sin (5 x) \frac{d}{d x} [\csc (5 x)]$

Step 4

Multiply $\sin (5 x)$ by $1$ .

$\sin (5 x) \frac{d}{d x} [\csc (5 x)]$

Step 5

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 $g (x) = 5 x$ .

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Step 5.1

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

$\sin (5 x) (\frac{d}{d u_{2}} [\csc (u_{2})] \frac{d}{d x} [5 x])$

Step 5.2

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

$\sin (5 x) (- \csc (u_{2}) \cot (u_{2}) \frac{d}{d x} [5 x])$

Step 5.3

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

$\sin (5 x) (- \csc (5 x) \cot (5 x) \frac{d}{d x} [5 x])$

Step 6

Differentiate.

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Step 6.1

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

$\sin (5 x) (- \csc (5 x) \cot (5 x) (5 \frac{d}{d x} [x]))$

Step 6.2

Multiply $5$ by $- 1$ .

$\sin (5 x) (- 5 \csc (5 x) \cot (5 x) \frac{d}{d x} [x])$

Step 6.3

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

$\sin (5 x) (- 5 \csc (5 x) \cot (5 x) \cdot 1)$

Step 6.4

Multiply $- 5$ by $1$ .

$\sin (5 x) (- 5 \csc (5 x) \cot (5 x))$

Step 7

Simplify.

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Step 7.1

Reorder the factors of $\sin (5 x) (- 5 \csc (5 x) \cot (5 x))$ .

$- 5 \cot (5 x) \csc (5 x) \sin (5 x)$

Step 7.2

Rewrite $\cot (5 x)$ in terms of sines and cosines.

$- 5 \frac{\cos (5 x)}{\sin (5 x)} \csc (5 x) \sin (5 x)$

Step 7.3

Combine $- 5$ and $\frac{\cos (5 x)}{\sin (5 x)}$ .

$\frac{- 5 \cos (5 x)}{\sin (5 x)} \csc (5 x) \sin (5 x)$

Step 7.4

Move the negative in front of the fraction.

$- \frac{5 \cos (5 x)}{\sin (5 x)} \csc (5 x) \sin (5 x)$

Step 7.5

Rewrite $\csc (5 x)$ in terms of sines and cosines.

$- \frac{5 \cos (5 x)}{\sin (5 x)} \cdot \frac{1}{\sin (5 x)} \sin (5 x)$

Step 7.6

Multiply $- \frac{5 \cos (5 x)}{\sin (5 x)} \cdot \frac{1}{\sin (5 x)}$ .

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Step 7.6.1

Multiply $\frac{1}{\sin (5 x)}$ by $\frac{5 \cos (5 x)}{\sin (5 x)}$ .

$- \frac{5 \cos (5 x)}{\sin (5 x) \sin (5 x)} \sin (5 x)$

Step 7.6.2

Raise $\sin (5 x)$ to the power of $1$ .

$- \frac{5 \cos (5 x)}{\sin^{1} (5 x) \sin (5 x)} \sin (5 x)$

Step 7.6.3

Raise $\sin (5 x)$ to the power of $1$ .

$- \frac{5 \cos (5 x)}{\sin^{1} (5 x) \sin^{1} (5 x)} \sin (5 x)$

Step 7.6.4

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

$- \frac{5 \cos (5 x)}{{\sin (5 x)}^{1 + 1}} \sin (5 x)$

Step 7.6.5

Add $1$ and $1$ .

$- \frac{5 \cos (5 x)}{\sin^{2} (5 x)} \sin (5 x)$

Step 7.7

Cancel the common factor of $\sin (5 x)$ .

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Step 7.7.1

Move the leading negative in $- \frac{5 \cos (5 x)}{\sin^{2} (5 x)}$ into the numerator.

$\frac{- 5 \cos (5 x)}{\sin^{2} (5 x)} \sin (5 x)$

Step 7.7.2

Factor $\sin (5 x)$ out of $\sin^{2} (5 x)$ .

$\frac{- 5 \cos (5 x)}{\sin (5 x) \sin (5 x)} \sin (5 x)$

Step 7.7.3

Cancel the common factor.

$\frac{- 5 \cos (5 x)}{\sin (5 x) \sin (5 x)} \sin (5 x)$

Step 7.7.4

Rewrite the expression.

$\frac{- 5 \cos (5 x)}{\sin (5 x)}$

Step 7.8

Move the negative in front of the fraction.

$- \frac{5 \cos (5 x)}{\sin (5 x)}$

Step 7.9

Separate fractions.

$- (\frac{5}{1} \cdot \frac{\cos (5 x)}{\sin (5 x)})$

Step 7.10

Convert from $\frac{\cos (5 x)}{\sin (5 x)}$ to $\cot (5 x)$ .

$- (\frac{5}{1} \cot (5 x))$

Step 7.11

Divide $5$ by $1$ .

$- (5 \cot (5 x))$

Step 7.12

Multiply $5$ by $- 1$ .

$- 5 \cot (5 x)$