Calculus Examples

$f (x) = 2 \ln (\csc (x^{2}))$

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Rewrite $\csc (x^{2})$ in terms of sines and cosines.

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

Step 4

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x^{2})}$ .

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

Step 5

Multiply $\sin (x^{2})$ by $1$ .

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

Step 6

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 $g (x) = x^{2}$ .

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

Differentiate using the Power Rule.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

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

$- 2 \sin (x^{2}) \csc (x^{2}) \cot (x^{2}) (2 x)$

Step 7.3

Multiply $2$ by $- 2$ .

$- 4 \sin (x^{2}) \csc (x^{2}) \cot (x^{2}) x$

Step 8

Simplify.

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Step 8.1

Reorder the factors of $- 4 \sin (x^{2}) \csc (x^{2}) \cot (x^{2}) x$ .

$- 4 x \cot (x^{2}) \csc (x^{2}) \sin (x^{2})$

Step 8.2

Rewrite $\cot (x^{2})$ in terms of sines and cosines.

$- 4 x \frac{\cos (x^{2})}{\sin (x^{2})} \csc (x^{2}) \sin (x^{2})$

Step 8.3

Multiply $- 4 x \frac{\cos (x^{2})}{\sin (x^{2})}$ .

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Step 8.3.1

Combine $- 4$ and $\frac{\cos (x^{2})}{\sin (x^{2})}$ .

$x \frac{- 4 \cos (x^{2})}{\sin (x^{2})} \csc (x^{2}) \sin (x^{2})$

Step 8.3.2

Combine $x$ and $\frac{- 4 \cos (x^{2})}{\sin (x^{2})}$ .

$\frac{x (- 4 \cos (x^{2}))}{\sin (x^{2})} \csc (x^{2}) \sin (x^{2})$

Step 8.4

Move $- 4$ to the left of $x$ .

$\frac{- 4 \cdot x \cos (x^{2})}{\sin (x^{2})} \csc (x^{2}) \sin (x^{2})$

Step 8.5

Move the negative in front of the fraction.

$- \frac{4 \cdot x \cos (x^{2})}{\sin (x^{2})} \csc (x^{2}) \sin (x^{2})$

Step 8.6

Rewrite $\csc (x^{2})$ in terms of sines and cosines.

$- \frac{4 x \cos (x^{2})}{\sin (x^{2})} \cdot \frac{1}{\sin (x^{2})} \sin (x^{2})$

Step 8.7

Multiply $- \frac{4 x \cos (x^{2})}{\sin (x^{2})} \cdot \frac{1}{\sin (x^{2})}$ .

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Step 8.7.1

Multiply $\frac{1}{\sin (x^{2})}$ by $\frac{4 x \cos (x^{2})}{\sin (x^{2})}$ .

$- \frac{4 x \cos (x^{2})}{\sin (x^{2}) \sin (x^{2})} \sin (x^{2})$

Step 8.7.2

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

$- \frac{4 x \cos (x^{2})}{\sin^{1} (x^{2}) \sin (x^{2})} \sin (x^{2})$

Step 8.7.3

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

$- \frac{4 x \cos (x^{2})}{\sin^{1} (x^{2}) \sin^{1} (x^{2})} \sin (x^{2})$

Step 8.7.4

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

$- \frac{4 x \cos (x^{2})}{{\sin (x^{2})}^{1 + 1}} \sin (x^{2})$

Step 8.7.5

Add $1$ and $1$ .

$- \frac{4 x \cos (x^{2})}{\sin^{2} (x^{2})} \sin (x^{2})$

Step 8.8

Cancel the common factor of $\sin (x^{2})$ .

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Step 8.8.1

Move the leading negative in $- \frac{4 x \cos (x^{2})}{\sin^{2} (x^{2})}$ into the numerator.

$\frac{- 4 x \cos (x^{2})}{\sin^{2} (x^{2})} \sin (x^{2})$

Step 8.8.2

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

$\frac{- 4 x \cos (x^{2})}{\sin (x^{2}) \sin (x^{2})} \sin (x^{2})$

Step 8.8.3

Cancel the common factor.

$\frac{- 4 x \cos (x^{2})}{\sin (x^{2}) \sin (x^{2})} \sin (x^{2})$

Step 8.8.4

Rewrite the expression.

$\frac{- 4 x \cos (x^{2})}{\sin (x^{2})}$

Step 8.9

Move the negative in front of the fraction.

$- \frac{4 x \cos (x^{2})}{\sin (x^{2})}$

Step 8.10

Separate fractions.

$- (\frac{4 x}{1} \cdot \frac{\cos (x^{2})}{\sin (x^{2})})$

Step 8.11

Convert from $\frac{\cos (x^{2})}{\sin (x^{2})}$ to $\cot (x^{2})$ .

$- (\frac{4 x}{1} \cot (x^{2}))$

Step 8.12

Divide $4 x$ by $1$ .

$- (4 x \cot (x^{2}))$

Step 8.13

Multiply $4$ by $- 1$ .

$- 4 x \cot (x^{2})$