Calculus Examples

$y = \frac{x}{e^{x}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{x}{e^{x}})$

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

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

Step 3.2

Differentiate using the Power Rule.

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Step 3.2.1

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

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Step 3.2.1.1

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

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

Step 3.2.1.2

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

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

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

Step 3.4

Simplify.

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Step 3.4.1

Reorder terms.

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

Step 3.4.2

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

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Step 3.4.2.1

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

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

Step 3.4.2.2

Multiply by $1$ .

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

Step 3.4.2.3

Factor $e^{x}$ out of $e^{x} (- x) + e^{x} \cdot 1$ .

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

Step 3.4.3

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

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Step 3.4.3.1

Factor $e^{2 x}$ out of $e^{x} (- x + 1)$ .

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

Step 3.4.3.2

Cancel the common factors.

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Step 3.4.3.2.1

Multiply by $1$ .

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

Step 3.4.3.2.2

Cancel the common factor.

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

Step 3.4.3.2.3

Rewrite the expression.

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

Step 3.4.3.2.4

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

$e^{- x} (- x + 1)$

Step 3.4.4

Apply the distributive property.

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

Step 3.4.5

Rewrite using the commutative property of multiplication.

$- e^{- x} x + e^{- x} \cdot 1$

Step 3.4.6

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

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

Step 3.4.7

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

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

Step 4

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

$y' = - x e^{- x} + e^{- x}$

Step 5

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

$\frac{d y}{d x} = - x e^{- x} + e^{- x}$