Calculus Examples

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

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{\cos (2 x)}{\sin^{2} (x)})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

Multiply the exponents in ${(\sin^{2} (x))}^{2}$ .

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Step 3.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sin^{2} (x) \frac{d}{d x} [\cos (2 x)] - \cos (2 x) \frac{d}{d x} [\sin^{2} (x)]}{{\sin (x)}^{2 \cdot 2}}$

Step 3.2.2

Multiply $2$ by $2$ .

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

Step 3.3

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

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Step 3.3.1

To apply the Chain Rule, set $u_{1}$ as $2 x$ .

$\frac{\sin^{2} (x) (\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d x} [2 x]) - \cos (2 x) \frac{d}{d x} [\sin^{2} (x)]}{\sin^{4} (x)}$

Step 3.3.2

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

$\frac{\sin^{2} (x) (- \sin (u_{1}) \frac{d}{d x} [2 x]) - \cos (2 x) \frac{d}{d x} [\sin^{2} (x)]}{\sin^{4} (x)}$

Step 3.3.3

Replace all occurrences of $u_{1}$ with $2 x$ .

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

Step 3.4

Differentiate.

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Step 3.4.1

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

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

Step 3.4.2

Multiply $2$ by $- 1$ .

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

Step 3.4.3

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

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

Step 3.4.4

Multiply $- 2$ by $1$ .

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

Step 3.5

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

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Step 3.5.1

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

$\frac{\sin^{2} (x) (- 2 \sin (2 x)) - \cos (2 x) (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [\sin (x)])}{\sin^{4} (x)}$

Step 3.5.2

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

$\frac{\sin^{2} (x) (- 2 \sin (2 x)) - \cos (2 x) (2 u_{2} \frac{d}{d x} [\sin (x)])}{\sin^{4} (x)}$

Step 3.5.3

Replace all occurrences of $u_{2}$ with $\sin (x)$ .

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

Step 3.6

Multiply $2$ by $- 1$ .

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

Step 3.7

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

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

Step 3.8

Simplify.

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Step 3.8.1

Rewrite using the commutative property of multiplication.

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

Step 3.8.2

Reorder terms.

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

Step 3.8.3

Simplify the numerator.

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Step 3.8.3.1

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

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Step 3.8.3.1.1

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

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

Step 3.8.3.1.2

Factor $2 \sin (x)$ out of $- 2 \cos (x) \cos (2 x) \sin (x)$ .

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

Step 3.8.3.1.3

Factor $2 \sin (x)$ out of $2 \sin (x) (- \sin (x) \sin (2 x)) + 2 \sin (x) (- \cos (x) \cos (2 x))$ .

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

Step 3.8.3.2

Factor $- 1$ out of $- \sin (x) \sin (2 x) - \cos (x) \cos (2 x)$ .

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Step 3.8.3.2.1

Factor $- 1$ out of $- \sin (x) \sin (2 x)$ .

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

Step 3.8.3.2.2

Factor $- 1$ out of $- \cos (x) \cos (2 x)$ .

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

Step 3.8.3.2.3

Factor $- 1$ out of $- (\sin (x) \sin (2 x)) - (\cos (x) \cos (2 x))$ .

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

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

Step 3.8.3.3

Combine exponents.

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Step 3.8.3.3.1

Factor out negative.

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

Step 3.8.3.3.2

Multiply $2$ by $- 1$ .

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

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

Step 3.8.4

Cancel the common factor of $\sin (x)$ and $\sin^{4} (x)$ .

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Step 3.8.4.1

Factor $\sin (x)$ out of $- 2 \sin (x) (\sin (x) \sin (2 x) + \cos (x) \cos (2 x))$ .

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

Step 3.8.4.2

Cancel the common factors.

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Step 3.8.4.2.1

Factor $\sin (x)$ out of $\sin^{4} (x)$ .

$\frac{\sin (x) (- 2 (\sin (x) \sin (2 x) + \cos (x) \cos (2 x)))}{\sin (x) \sin^{3} (x)}$

Step 3.8.4.2.2

Cancel the common factor.

$\frac{\sin (x) (- 2 (\sin (x) \sin (2 x) + \cos (x) \cos (2 x)))}{\sin (x) \sin^{3} (x)}$

Step 3.8.4.2.3

Rewrite the expression.

$\frac{- 2 (\sin (x) \sin (2 x) + \cos (x) \cos (2 x))}{\sin^{3} (x)}$

Step 3.8.5

Simplify the numerator.

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Step 3.8.5.1

Simplify each term.

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Step 3.8.5.1.1

Apply the sine double-angle identity.

$\frac{- 2 (\sin (x) (2 \sin (x) \cos (x)) + \cos (x) \cos (2 x))}{\sin^{3} (x)}$

Step 3.8.5.1.2

Rewrite using the commutative property of multiplication.

$\frac{- 2 (2 \sin (x) \sin (x) \cos (x) + \cos (x) \cos (2 x))}{\sin^{3} (x)}$

Step 3.8.5.1.3

Multiply $2 \sin (x) \sin (x)$ .

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Step 3.8.5.1.3.1

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

$\frac{- 2 (2 (\sin^{1} (x) \sin (x)) \cos (x) + \cos (x) \cos (2 x))}{\sin^{3} (x)}$

Step 3.8.5.1.3.2

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

$\frac{- 2 (2 (\sin^{1} (x) \sin^{1} (x)) \cos (x) + \cos (x) \cos (2 x))}{\sin^{3} (x)}$

Step 3.8.5.1.3.3

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

$\frac{- 2 (2 {\sin (x)}^{1 + 1} \cos (x) + \cos (x) \cos (2 x))}{\sin^{3} (x)}$

Step 3.8.5.1.3.4

Add $1$ and $1$ .

$\frac{- 2 (2 \sin^{2} (x) \cos (x) + \cos (x) \cos (2 x))}{\sin^{3} (x)}$

Step 3.8.5.1.4

Use the double-angle identity to transform $\cos (2 x)$ to $1 - 2 \sin^{2} (x)$ .

$\frac{- 2 (2 \sin^{2} (x) \cos (x) + \cos (x) (1 - 2 \sin^{2} (x)))}{\sin^{3} (x)}$

Step 3.8.5.1.5

Apply the distributive property.

$\frac{- 2 (2 \sin^{2} (x) \cos (x) + \cos (x) \cdot 1 + \cos (x) (- 2 \sin^{2} (x)))}{\sin^{3} (x)}$

Step 3.8.5.1.6

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

$\frac{- 2 (2 \sin^{2} (x) \cos (x) + \cos (x) + \cos (x) (- 2 \sin^{2} (x)))}{\sin^{3} (x)}$

Step 3.8.5.1.7

Rewrite using the commutative property of multiplication.

$\frac{- 2 (2 \sin^{2} (x) \cos (x) + \cos (x) - 2 \cos (x) \sin^{2} (x))}{\sin^{3} (x)}$

Step 3.8.5.2

Combine the opposite terms in $2 \sin^{2} (x) \cos (x) + \cos (x) - 2 \cos (x) \sin^{2} (x)$ .

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Step 3.8.5.2.1

Reorder the factors in the terms $2 \sin^{2} (x) \cos (x)$ and $- 2 \cos (x) \sin^{2} (x)$ .

$\frac{- 2 (2 \sin^{2} (x) \cos (x) + \cos (x) - 2 \sin^{2} (x) \cos (x))}{\sin^{3} (x)}$

Step 3.8.5.2.2

Subtract $2 \sin^{2} (x) \cos (x)$ from $2 \sin^{2} (x) \cos (x)$ .

$\frac{- 2 (0 + \cos (x))}{\sin^{3} (x)}$

Step 3.8.5.2.3

Add $0$ and $\cos (x)$ .

$\frac{- 2 \cos (x)}{\sin^{3} (x)}$

Step 3.8.6

Factor $\sin (x)$ out of $\sin^{3} (x)$ .

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

Step 3.8.7

Separate fractions.

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

Step 3.8.8

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

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

Step 3.8.9

Multiply by $1$ .

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

Step 3.8.10

Separate fractions.

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

Step 3.8.11

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

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

Step 3.8.12

Divide $- 2$ by $1$ .

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

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = - 2 \csc^{2} (x) \cot (x)$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - 2 \csc^{2} (x) \cot (x)$