Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Quotient Rule which states that $\frac{d}{d y} [\frac{f (y)}{g (y)}]$ is $\frac{g (y) \frac{d}{d y} [f (y)] - f (y) \frac{d}{d y} [g (y)]}{{g (y)}^{2}}$ where $f (y) = x$ and $g (y) = e^{x}$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.3

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

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

Step 3.4

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = e^{y}$ and $g (y) = x$ .

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Step 3.4.1

To apply the Chain Rule, set $u$ as $x$ .

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

Step 3.4.2

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

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

Step 3.4.3

Replace all occurrences of $u$ with $x$ .

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

Step 3.5

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

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

Step 3.6

Simplify.

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Step 3.6.1

Reorder terms.

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

Step 3.6.2

Simplify the numerator.

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Step 3.6.2.1

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

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Step 3.6.2.1.1

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

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

Step 3.6.2.1.2

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

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

Step 3.6.2.1.3

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

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

Step 3.6.2.2

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

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Step 3.6.2.2.1

Multiply by $1$ .

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

Step 3.6.2.2.2

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

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

Step 3.6.2.2.3

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

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

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

Step 3.6.3

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

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Step 3.6.3.1

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

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

Step 3.6.3.2

Cancel the common factors.

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Step 3.6.3.2.1

Multiply by $1$ .

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

Step 3.6.3.2.2

Cancel the common factor.

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

Step 3.6.3.2.3

Rewrite the expression.

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

Step 3.6.3.2.4

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

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

Step 3.6.4

Apply the distributive property.

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

Step 3.6.5

Multiply $x'$ by $1$ .

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

Step 3.6.6

Rewrite using the commutative property of multiplication.

$x' e^{- x} - x' e^{- x} x$

Step 3.6.7

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

$e^{- x} x' - x e^{- x} x'$

Step 4

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

$1 = e^{- x} x' - x e^{- x} x'$

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $e^{- x} x' - x e^{- x} x' = 1$ .

$e^{- x} x' - x e^{- x} x' = 1$

Step 5.2

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

$x' e^{- x} - x x' e^{- x} = 1$

Step 5.3

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

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Step 5.3.1

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

$x' e^{- x} \cdot 1 - x x' e^{- x} = 1$

Step 5.3.2

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

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

Step 5.3.3

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

$x' e^{- x} (1 - x) = 1$

Step 5.4

Divide each term in $x' e^{- x} (1 - x) = 1$ by $e^{- x} (1 - x)$ and simplify.

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Step 5.4.1

Divide each term in $x' e^{- x} (1 - x) = 1$ by $e^{- x} (1 - x)$ .

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

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

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

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Step 5.4.2.1.1

Cancel the common factor.

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

Step 5.4.2.1.2

Rewrite the expression.

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

Step 5.4.2.2

Cancel the common factor of $1 - x$ .

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Step 5.4.2.2.1

Cancel the common factor.

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

Step 5.4.2.2.2

Divide $x'$ by $1$ .

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

Step 6

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

$\frac{d x}{d y} = \frac{1}{e^{- x} (1 - x)}$