Calculus Examples

$y = 4 \sin (x) \csc (x^{2})$

Step 1

Since $4$ is constant with respect to $x$ , the derivative of $4 \sin (x) \csc (x^{2})$ with respect to $x$ is $4 \frac{d}{d x} [\sin (x) \csc (x^{2})]$ .

$4 \frac{d}{d x} [\sin (x) \csc (x^{2})]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \sin (x)$ and $g (x) = \csc (x^{2})$ .

$4 (\sin (x) \frac{d}{d x} [\csc (x^{2})] + \csc (x^{2}) \frac{d}{d x} [\sin (x)])$

Step 3

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 3.1

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

$4 (\sin (x) (\frac{d}{d u} [\csc (u)] \frac{d}{d x} [x^{2}]) + \csc (x^{2}) \frac{d}{d x} [\sin (x)])$

Step 3.2

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

$4 (\sin (x) (- \csc (u) \cot (u) \frac{d}{d x} [x^{2}]) + \csc (x^{2}) \frac{d}{d x} [\sin (x)])$

Step 3.3

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

$4 (\sin (x) (- \csc (x^{2}) \cot (x^{2}) \frac{d}{d x} [x^{2}]) + \csc (x^{2}) \frac{d}{d x} [\sin (x)])$

Step 4

Differentiate using the Power Rule.

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Step 4.1

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

$4 (\sin (x) (- \csc (x^{2}) \cot (x^{2}) (2 x)) + \csc (x^{2}) \frac{d}{d x} [\sin (x)])$

Step 4.2

Multiply $2$ by $- 1$ .

$4 (\sin (x) (- 2 \csc (x^{2}) \cot (x^{2}) x) + \csc (x^{2}) \frac{d}{d x} [\sin (x)])$

Step 5

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $- 2$ by $4$ .

$- 8 \sin (x) \csc (x^{2}) \cot (x^{2}) x + 4 \csc (x^{2}) \cos (x)$

Step 6.3

Reorder terms.

$- 8 x \cot (x^{2}) \csc (x^{2}) \sin (x) + 4 \cos (x) \csc (x^{2})$

Step 6.4

Simplify each term.

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Step 6.4.1

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

$- 8 x \frac{\cos (x^{2})}{\sin (x^{2})} \csc (x^{2}) \sin (x) + 4 \cos (x) \csc (x^{2})$

Step 6.4.2

Multiply $- 8 x \frac{\cos (x^{2})}{\sin (x^{2})}$ .

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Step 6.4.2.1

Combine $- 8$ and $\frac{\cos (x^{2})}{\sin (x^{2})}$ .

$x \frac{- 8 \cos (x^{2})}{\sin (x^{2})} \csc (x^{2}) \sin (x) + 4 \cos (x) \csc (x^{2})$

Step 6.4.2.2

Combine $x$ and $\frac{- 8 \cos (x^{2})}{\sin (x^{2})}$ .

$\frac{x (- 8 \cos (x^{2}))}{\sin (x^{2})} \csc (x^{2}) \sin (x) + 4 \cos (x) \csc (x^{2})$

Step 6.4.3

Move $- 8$ to the left of $x$ .

$\frac{- 8 \cdot x \cos (x^{2})}{\sin (x^{2})} \csc (x^{2}) \sin (x) + 4 \cos (x) \csc (x^{2})$

Step 6.4.4

Move the negative in front of the fraction.

$- \frac{8 \cdot x \cos (x^{2})}{\sin (x^{2})} \csc (x^{2}) \sin (x) + 4 \cos (x) \csc (x^{2})$

Step 6.4.5

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

$- \frac{8 x \cos (x^{2})}{\sin (x^{2})} \cdot \frac{1}{\sin (x^{2})} \sin (x) + 4 \cos (x) \csc (x^{2})$

Step 6.4.6

Multiply $- \frac{8 x \cos (x^{2})}{\sin (x^{2})} \cdot \frac{1}{\sin (x^{2})}$ .

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Step 6.4.6.1

Multiply $\frac{1}{\sin (x^{2})}$ by $\frac{8 x \cos (x^{2})}{\sin (x^{2})}$ .

$- \frac{8 x \cos (x^{2})}{\sin (x^{2}) \sin (x^{2})} \sin (x) + 4 \cos (x) \csc (x^{2})$

Step 6.4.6.2

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

$- \frac{8 x \cos (x^{2})}{\sin^{1} (x^{2}) \sin (x^{2})} \sin (x) + 4 \cos (x) \csc (x^{2})$

Step 6.4.6.3

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

$- \frac{8 x \cos (x^{2})}{\sin^{1} (x^{2}) \sin^{1} (x^{2})} \sin (x) + 4 \cos (x) \csc (x^{2})$

Step 6.4.6.4

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

$- \frac{8 x \cos (x^{2})}{{\sin (x^{2})}^{1 + 1}} \sin (x) + 4 \cos (x) \csc (x^{2})$

Step 6.4.6.5

Add $1$ and $1$ .

$- \frac{8 x \cos (x^{2})}{\sin^{2} (x^{2})} \sin (x) + 4 \cos (x) \csc (x^{2})$

Step 6.4.7

Combine $\sin (x)$ and $\frac{8 x \cos (x^{2})}{\sin^{2} (x^{2})}$ .

$- \frac{\sin (x) (8 x \cos (x^{2}))}{\sin^{2} (x^{2})} + 4 \cos (x) \csc (x^{2})$

Step 6.4.8

Move $8$ to the left of $\sin (x)$ .

$- \frac{8 \cdot \sin (x) x \cos (x^{2})}{\sin^{2} (x^{2})} + 4 \cos (x) \csc (x^{2})$

Step 6.4.9

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

$- \frac{8 \sin (x) x \cos (x^{2})}{\sin^{2} (x^{2})} + 4 \cos (x) \frac{1}{\sin (x^{2})}$

Step 6.4.10

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

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Step 6.4.10.1

Combine $\frac{1}{\sin (x^{2})}$ and $4$ .

$- \frac{8 \sin (x) x \cos (x^{2})}{\sin^{2} (x^{2})} + \frac{4}{\sin (x^{2})} \cos (x)$

Step 6.4.10.2

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

$- \frac{8 \sin (x) x \cos (x^{2})}{\sin^{2} (x^{2})} + \frac{4 \cos (x)}{\sin (x^{2})}$

Step 6.5

Simplify each term.

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Step 6.5.1

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

$- \frac{8 \sin (x) x \cos (x^{2})}{\sin (x^{2}) \sin (x^{2})} + \frac{4 \cos (x)}{\sin (x^{2})}$

Step 6.5.2

Separate fractions.

$- (\frac{8 x \cos (x^{2})}{\sin (x^{2})} \cdot \frac{\sin (x)}{\sin (x^{2})}) + \frac{4 \cos (x)}{\sin (x^{2})}$

Step 6.5.3

Rewrite $\frac{\sin (x)}{\sin (x^{2})}$ as a product.

$- (\frac{8 x \cos (x^{2})}{\sin (x^{2})} (\sin (x) \frac{1}{\sin (x^{2})})) + \frac{4 \cos (x)}{\sin (x^{2})}$

Step 6.5.4

Write $\sin (x)$ as a fraction with denominator $1$ .

$- (\frac{8 x \cos (x^{2})}{\sin (x^{2})} (\frac{\sin (x)}{1} \cdot \frac{1}{\sin (x^{2})})) + \frac{4 \cos (x)}{\sin (x^{2})}$

Step 6.5.5

Simplify.

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Step 6.5.5.1

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

$- (\frac{8 x \cos (x^{2})}{\sin (x^{2})} (\sin (x) \frac{1}{\sin (x^{2})})) + \frac{4 \cos (x)}{\sin (x^{2})}$

Step 6.5.5.2

Convert from $\frac{1}{\sin (x^{2})}$ to $\csc (x^{2})$ .

$- (\frac{8 x \cos (x^{2})}{\sin (x^{2})} (\sin (x) \csc (x^{2}))) + \frac{4 \cos (x)}{\sin (x^{2})}$

Step 6.5.6

Separate fractions.

$- (\frac{8 x}{1} \cdot \frac{\cos (x^{2})}{\sin (x^{2})} (\sin (x) \csc (x^{2}))) + \frac{4 \cos (x)}{\sin (x^{2})}$

Step 6.5.7

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

$- (\frac{8 x}{1} \cot (x^{2}) (\sin (x) \csc (x^{2}))) + \frac{4 \cos (x)}{\sin (x^{2})}$

Step 6.5.8

Divide $8 x$ by $1$ .

$- (8 x \cot (x^{2}) (\sin (x) \csc (x^{2}))) + \frac{4 \cos (x)}{\sin (x^{2})}$

Step 6.5.9

Multiply $8$ by $- 1$ .

$- 8 (x \cot (x^{2}) \sin (x) \csc (x^{2})) + \frac{4 \cos (x)}{\sin (x^{2})}$

Step 6.5.10

Separate fractions.

$- 8 x \cot (x^{2}) \sin (x) \csc (x^{2}) + \frac{4}{1} \cdot \frac{\cos (x)}{\sin (x^{2})}$

Step 6.5.11

Rewrite $\frac{\cos (x)}{\sin (x^{2})}$ as a product.

$- 8 x \cot (x^{2}) \sin (x) \csc (x^{2}) + \frac{4}{1} (\cos (x) \frac{1}{\sin (x^{2})})$

Step 6.5.12

Write $\cos (x)$ as a fraction with denominator $1$ .

$- 8 x \cot (x^{2}) \sin (x) \csc (x^{2}) + \frac{4}{1} (\frac{\cos (x)}{1} \cdot \frac{1}{\sin (x^{2})})$

Step 6.5.13

Simplify.

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Step 6.5.13.1

Divide $\cos (x)$ by $1$ .

$- 8 x \cot (x^{2}) \sin (x) \csc (x^{2}) + \frac{4}{1} (\cos (x) \frac{1}{\sin (x^{2})})$

Step 6.5.13.2

Convert from $\frac{1}{\sin (x^{2})}$ to $\csc (x^{2})$ .

$- 8 x \cot (x^{2}) \sin (x) \csc (x^{2}) + \frac{4}{1} (\cos (x) \csc (x^{2}))$

Step 6.5.14

Divide $4$ by $1$ .

$- 8 x \cot (x^{2}) \sin (x) \csc (x^{2}) + 4 \cos (x) \csc (x^{2})$