Calculus Examples

$y = \ln (\csc (x))$

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 $g (x) = \csc (x)$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\csc (x)$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\csc (x)]$

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $\csc (x)$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

The derivative of $\csc (x)$ with respect to $x$ is $- \csc (x) \cot (x)$ .

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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Step 6.4.1

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

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

Step 6.4.2

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

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

Step 6.4.3

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

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

Step 6.4.4

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

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

Step 6.4.5

Add $1$ and $1$ .

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

Step 6.5

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

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Step 6.5.1

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

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

Step 6.5.2

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

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

Step 6.5.3

Cancel the common factor.

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

Step 6.5.4

Rewrite the expression.

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

Step 6.6

Move the negative in front of the fraction.

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

Step 6.7

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

$- \cot (x)$