Calculus Examples

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

Step 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 $f (x) = \sin (x) + \cos (x)$ and $g (x) = \sin (x) - \cos (x)$ .

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

Step 2

By the Sum Rule, the derivative of $\sin (x) + \cos (x)$ with respect to $x$ is $\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [\cos (x)]$ .

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

Step 3

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

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

Step 4

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

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

Step 5

By the Sum Rule, the derivative of $\sin (x) - \cos (x)$ with respect to $x$ is $\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [- \cos (x)]$ .

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

Step 6

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

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

Step 7

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cos (x)$ with respect to $x$ is $- \frac{d}{d x} [\cos (x)]$ .

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

Step 8

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

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

Step 9

Multiply.

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Step 9.1

Multiply $- 1$ by $- 1$ .

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

Step 9.2

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

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

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

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

Step 10.2

Simplify the numerator.

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Step 10.2.1

Simplify each term.

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Step 10.2.1.1

Expand $(\sin (x) - \cos (x)) (\cos (x) - \sin (x))$ using the FOIL Method.

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Step 10.2.1.1.1

Apply the distributive property.

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

Step 10.2.1.1.2

Apply the distributive property.

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

Step 10.2.1.1.3

Apply the distributive property.

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

Step 10.2.1.2

Simplify and combine like terms.

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Step 10.2.1.2.1

Simplify each term.

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Step 10.2.1.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 10.2.1.2.1.2

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

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Step 10.2.1.2.1.2.1

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

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

Step 10.2.1.2.1.2.2

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

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

Step 10.2.1.2.1.2.3

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

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

Step 10.2.1.2.1.2.4

Add $1$ and $1$ .

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

Step 10.2.1.2.1.3

Multiply $- \cos (x) \cos (x)$ .

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Step 10.2.1.2.1.3.1

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

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

Step 10.2.1.2.1.3.2

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

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

Step 10.2.1.2.1.3.3

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

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

Step 10.2.1.2.1.3.4

Add $1$ and $1$ .

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

Step 10.2.1.2.1.4

Multiply $- \cos (x) (- \sin (x))$ .

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Step 10.2.1.2.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 10.2.1.2.1.4.2

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

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

Step 10.2.1.2.2

Reorder the factors of $\sin (x) \cos (x)$ .

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

Step 10.2.1.2.3

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

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

Step 10.2.1.3

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

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

Step 10.2.1.4

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

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

Step 10.2.1.5

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

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

Step 10.2.1.6

Apply pythagorean identity.

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

Step 10.2.1.7

Simplify each term.

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Step 10.2.1.7.1

Reorder $2 \cos (x)$ and $\sin (x)$ .

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

Step 10.2.1.7.2

Reorder $\sin (x)$ and $2$ .

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

Step 10.2.1.7.3

Apply the sine double-angle identity.

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

Step 10.2.1.7.4

Multiply $- 1$ by $1$ .

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

Step 10.2.1.8

Expand $(- \sin (x) - \cos (x)) (\cos (x) + \sin (x))$ using the FOIL Method.

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Step 10.2.1.8.1

Apply the distributive property.

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

Step 10.2.1.8.2

Apply the distributive property.

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

Step 10.2.1.8.3

Apply the distributive property.

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

Step 10.2.1.9

Simplify and combine like terms.

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Step 10.2.1.9.1

Simplify each term.

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Step 10.2.1.9.1.1

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

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Step 10.2.1.9.1.1.1

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

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

Step 10.2.1.9.1.1.2

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

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

Step 10.2.1.9.1.1.3

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

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

Step 10.2.1.9.1.1.4

Add $1$ and $1$ .

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

Step 10.2.1.9.1.2

Multiply $- \cos (x) \cos (x)$ .

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Step 10.2.1.9.1.2.1

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

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

Step 10.2.1.9.1.2.2

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

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

Step 10.2.1.9.1.2.3

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

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

Step 10.2.1.9.1.2.4

Add $1$ and $1$ .

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

Step 10.2.1.9.2

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

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

Step 10.2.1.9.3

Subtract $\cos (x) \sin (x)$ from $- \cos (x) \sin (x)$ .

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

Step 10.2.1.10

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

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

Step 10.2.1.11

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

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

Step 10.2.1.12

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

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

Step 10.2.1.13

Apply pythagorean identity.

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

Step 10.2.1.14

Multiply $- 1$ by $1$ .

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

Step 10.2.2

Subtract $1$ from $- 1$ .

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

Step 10.3

Reorder terms.

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

Step 10.4

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

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

Step 10.5

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 10.6

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

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

Step 10.7

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

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

Step 10.8

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

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

Step 10.9

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

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

Step 10.10

Move the negative in front of the fraction.

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