Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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 $f (x) = \cos (x)$ and $g (x) = \sin^{2} (x)$ .

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.2.2

Multiply $2$ by $2$ .

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

Step 3.3

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

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

Step 3.4

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

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Step 3.4.1

Move $\sin (x)$ .

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

Step 3.4.2

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

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Step 3.4.2.1

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

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

Step 3.4.2.2

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

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

Step 3.4.3

Add $1$ and $2$ .

$\frac{\sin^{3} (x) \cdot - 1 - \cos (x) \frac{d}{d x} [\sin^{2} (x)]}{\sin^{4} (x)}$

Step 3.5

Simplify the expression.

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Step 3.5.1

Move $- 1$ to the left of $\sin^{3} (x)$ .

$\frac{- 1 \cdot \sin^{3} (x) - \cos (x) \frac{d}{d x} [\sin^{2} (x)]}{\sin^{4} (x)}$

Step 3.5.2

Rewrite $- 1 \sin^{3} (x)$ as $- \sin^{3} (x)$ .

$\frac{- \sin^{3} (x) - \cos (x) \frac{d}{d x} [\sin^{2} (x)]}{\sin^{4} (x)}$

Step 3.6

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

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Step 3.6.1

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

$\frac{- \sin^{3} (x) - \cos (x) (\frac{d}{d u} [u^{2}] \frac{d}{d x} [\sin (x)])}{\sin^{4} (x)}$

Step 3.6.2

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

$\frac{- \sin^{3} (x) - \cos (x) (2 u \frac{d}{d x} [\sin (x)])}{\sin^{4} (x)}$

Step 3.6.3

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

$\frac{- \sin^{3} (x) - \cos (x) (2 \sin (x) \frac{d}{d x} [\sin (x)])}{\sin^{4} (x)}$

Step 3.7

Simplify with factoring out.

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Step 3.7.1

Multiply $2$ by $- 1$ .

$\frac{- \sin^{3} (x) - 2 \cos (x) (\sin (x) \frac{d}{d x} [\sin (x)])}{\sin^{4} (x)}$

Step 3.7.2

Factor $\sin (x)$ out of $- \sin^{3} (x) - 2 \cos (x) \sin (x) \frac{d}{d x} [\sin (x)]$ .

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Step 3.7.2.1

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

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

Step 3.7.2.2

Factor $\sin (x)$ out of $- 2 \cos (x) \sin (x) \frac{d}{d x} [\sin (x)]$ .

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

Step 3.7.2.3

Factor $\sin (x)$ out of $\sin (x) (- \sin^{2} (x)) + \sin (x) (- 2 \cos (x) \frac{d}{d x} [\sin (x)])$ .

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

Step 3.8

Cancel the common factors.

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Step 3.8.1

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

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

Step 3.8.2

Cancel the common factor.

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

Step 3.8.3

Rewrite the expression.

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

Step 3.9

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

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

Step 3.10

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

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

Step 3.11

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

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

Step 3.12

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

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

Step 3.13

Add $1$ and $1$ .

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

Step 3.14

Simplify.

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Step 3.14.1

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

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

Step 3.14.2

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

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

Step 3.14.3

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

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

Step 3.14.4

Rewrite $- (\sin^{2} (x) + 2 \cos^{2} (x))$ as $- 1 (\sin^{2} (x) + 2 \cos^{2} (x))$ .

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

Step 3.14.5

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

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

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