Calculus Examples

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

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

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

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

Step 5

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 6

Differentiate using the Sum Rule.

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Step 6.1

Multiply $- 1$ by $- 1$ .

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

Step 6.2

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

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

Step 6.3

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 7

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 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

Simplify.

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Step 9.1

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 9.2

Simplify the numerator.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

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

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Step 9.2.1.1.1

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 9.2.1.1.2

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 9.2.1.1.3

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 9.2.1.2

Simplify and combine like terms.

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Step 9.2.1.2.1

Simplify each term.

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Step 9.2.1.2.1.1

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

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Step 9.2.1.2.1.1.1

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 9.2.1.2.1.1.2

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 9.2.1.2.1.1.3

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 9.2.1.2.1.1.4

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 9.2.1.2.1.2

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

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Step 9.2.1.2.1.2.1

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 9.2.1.2.1.2.2

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 9.2.1.2.1.2.3

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

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

Step 9.2.1.2.1.2.4

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 9.2.1.2.2

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 9.2.1.2.3

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 9.2.1.3

Apply pythagorean identity.

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

Step 9.2.1.4

Simplify each term.

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Step 9.2.1.4.1

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

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

Step 9.2.1.4.2

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

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

Step 9.2.1.4.3

Apply the sine double-angle identity.

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

Step 9.2.1.5

Multiply $- - \cos (x)$ .

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Step 9.2.1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 9.2.1.5.2

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

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

Step 9.2.1.6

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

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Step 9.2.1.6.1

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 9.2.1.6.2

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 9.2.1.6.3

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 9.2.1.7

Simplify and combine like terms.

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Step 9.2.1.7.1

Simplify each term.

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Step 9.2.1.7.1.1

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

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Step 9.2.1.7.1.1.1

Multiply $- 1$ by $- 1$ .

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

Step 9.2.1.7.1.1.2

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

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

Step 9.2.1.7.1.1.3

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 9.2.1.7.1.1.4

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 9.2.1.7.1.1.5

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 9.2.1.7.1.1.6

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 9.2.1.7.1.2

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

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Step 9.2.1.7.1.2.1

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 9.2.1.7.1.2.2

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 9.2.1.7.1.2.3

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 9.2.1.7.1.2.4

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 9.2.1.7.1.3

Rewrite using the commutative property of multiplication.

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

Step 9.2.1.7.2

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 9.2.1.7.3

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 9.2.1.8

Apply pythagorean identity.

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

Step 9.2.2

Add $1$ and $1$ .

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

Step 9.3

Reorder terms.

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

Step 9.4

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 9.5

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

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

Step 9.6

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 9.7

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

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

Step 9.8

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 9.9

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 9.10

Move the negative in front of the fraction.

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