Calculus Examples

$\frac{x - 1}{e^{x}}$

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 $f (x) = x - 1$ and $g (x) = e^{x}$ .

$\frac{e^{x} \frac{d}{d x} [x - 1] - (x - 1) \frac{d}{d x} [e^{x}]}{{(e^{x})}^{2}}$

Step 2

Differentiate.

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

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

Step 2.1.2

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

$\frac{e^{x} \frac{d}{d x} [x - 1] - (x - 1) \frac{d}{d x} [e^{x}]}{e^{2 x}}$

Step 2.2

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

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

Step 2.3

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

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

Step 2.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$\frac{e^{x} (1 + 0) - (x - 1) \frac{d}{d x} [e^{x}]}{e^{2 x}}$

Step 2.5

Simplify the expression.

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Step 2.5.1

Add $1$ and $0$ .

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

Step 2.5.2

Multiply $e^{x}$ by $1$ .

$\frac{e^{x} - (x - 1) \frac{d}{d x} [e^{x}]}{e^{2 x}}$

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

$\frac{e^{x} - x e^{x} - - 1 e^{x}}{e^{2 x}}$

Step 4.3

Simplify the numerator.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply $- 1$ by $- 1$ .

$\frac{e^{x} - x e^{x} + 1 e^{x}}{e^{2 x}}$

Step 4.3.1.2

Multiply $e^{x}$ by $1$ .

$\frac{e^{x} - x e^{x} + e^{x}}{e^{2 x}}$

Step 4.3.2

Add $e^{x}$ and $e^{x}$ .

$\frac{- x e^{x} + 2 e^{x}}{e^{2 x}}$

Step 4.4

Reorder terms.

$\frac{- e^{x} x + 2 e^{x}}{e^{2 x}}$

Step 4.5

Factor $e^{x}$ out of $- e^{x} x + 2 e^{x}$ .

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Step 4.5.1

Factor $e^{x}$ out of $- e^{x} x$ .

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

Step 4.5.2

Factor $e^{x}$ out of $2 e^{x}$ .

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

Step 4.5.3

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

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

Step 4.6

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

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Step 4.6.1

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

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

Step 4.6.2

Cancel the common factors.

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Step 4.6.2.1

Multiply by $1$ .

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

Step 4.6.2.2

Cancel the common factor.

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

Step 4.6.2.3

Rewrite the expression.

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

Step 4.6.2.4

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

$e^{- x} (- x + 2)$

Step 4.7

Apply the distributive property.

$e^{- x} (- x) + e^{- x} \cdot 2$

Step 4.8

Rewrite using the commutative property of multiplication.

$- e^{- x} x + e^{- x} \cdot 2$

Step 4.9

Move $2$ to the left of $e^{- x}$ .

$- e^{- x} x + 2 \cdot e^{- x}$

Step 4.10

Reorder factors in $- e^{- x} x + 2 e^{- x}$ .

$- x e^{- x} + 2 e^{- x}$