Calculus Examples

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

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 $g (x) = e^{x}$ .

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

Step 2

Differentiate using the Sum Rule.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Move $2$ to the left of $x$ .

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

Step 2.2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Apply the distributive property.

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

Step 6.4

Simplify the numerator.

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Step 6.4.1

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

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Step 6.4.1.1

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

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

Step 6.4.1.2

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

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

Step 6.4.1.3

Add $e^{x} (- \sin (x)) - \sin (x) e^{x}$ and $0$ .

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

Step 6.4.2

Rewrite using the commutative property of multiplication.

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

Step 6.4.3

Subtract $\sin (x) e^{x}$ from $- e^{x} \sin (x)$ .

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Step 6.4.3.1

Move $\sin (x)$ .

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

Step 6.4.3.2

Subtract $e^{x} \sin (x)$ from $- e^{x} \sin (x)$ .

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

Step 6.5

Cancel the common factor of $e^{x}$ and $e^{2 x}$ .

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Step 6.5.1

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

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

Step 6.5.2

Cancel the common factors.

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Step 6.5.2.1

Multiply by $1$ .

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

Step 6.5.2.2

Cancel the common factor.

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

Step 6.5.2.3

Rewrite the expression.

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

Step 6.5.2.4

Divide $- 2 e^{- x} \sin (x)$ by $1$ .

$- 2 e^{- x} \sin (x)$