Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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Step 2.1.1

To apply the Chain Rule, set $u$ as $\frac{x}{y}$ .

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

Step 2.1.2

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

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

Step 2.1.3

Replace all occurrences of $u$ with $\frac{x}{y}$ .

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

Step 2.2

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 $g (x) = y$ .

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

Step 2.3

Differentiate using the Power Rule.

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Step 2.3.1

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

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

Step 2.3.2

Multiply $y$ by $1$ .

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

Step 2.4

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

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

Step 2.5

Combine $e^{\frac{x}{y}}$ and $\frac{y - x y'}{y^{2}}$ .

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

Step 3

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

$1$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Multiply both sides by $y^{2}$ .

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

Step 5.2

Simplify.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

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

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Step 5.2.1.1.1

Cancel the common factor of $y^{2}$ .

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Step 5.2.1.1.1.1

Cancel the common factor.

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

Step 5.2.1.1.1.2

Rewrite the expression.

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

Step 5.2.1.1.2

Apply the distributive property.

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

Step 5.2.1.1.3

Simplify the expression.

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Step 5.2.1.1.3.1

Rewrite using the commutative property of multiplication.

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

Step 5.2.1.1.3.2

Reorder factors in $e^{\frac{x}{y}} y - e^{\frac{x}{y}} (x y')$ .

$y e^{\frac{x}{y}} - x y' e^{\frac{x}{y}} = 1 y^{2}$

Step 5.2.1.1.3.3

Move $x$ .

$y e^{\frac{x}{y}} - y' x e^{\frac{x}{y}} = 1 y^{2}$

Step 5.2.1.1.3.4

Reorder $y e^{\frac{x}{y}}$ and $- y' x e^{\frac{x}{y}}$ .

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Multiply $y^{2}$ by $1$ .

$- y' x e^{\frac{x}{y}} + y e^{\frac{x}{y}} = y^{2}$

Step 5.3

Solve for $y'$ .

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Step 5.3.1

Subtract $y e^{\frac{x}{y}}$ from both sides of the equation.

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

Step 5.3.2

Divide each term in $- y' x e^{\frac{x}{y}} = y^{2} - y e^{\frac{x}{y}}$ by $- x e^{\frac{x}{y}}$ and simplify.

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Step 5.3.2.1

Divide each term in $- y' x e^{\frac{x}{y}} = y^{2} - y e^{\frac{x}{y}}$ by $- x e^{\frac{x}{y}}$ .

$\frac{- y' x e^{\frac{x}{y}}}{- x e^{\frac{x}{y}}} = \frac{y^{2}}{- x e^{\frac{x}{y}}} + \frac{- y e^{\frac{x}{y}}}{- x e^{\frac{x}{y}}}$

Step 5.3.2.2

Simplify the left side.

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Step 5.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{y' x e^{\frac{x}{y}}}{x e^{\frac{x}{y}}} = \frac{y^{2}}{- x e^{\frac{x}{y}}} + \frac{- y e^{\frac{x}{y}}}{- x e^{\frac{x}{y}}}$

Step 5.3.2.2.2

Cancel the common factor of $x$ .

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Step 5.3.2.2.2.1

Cancel the common factor.

$\frac{y' x e^{\frac{x}{y}}}{x e^{\frac{x}{y}}} = \frac{y^{2}}{- x e^{\frac{x}{y}}} + \frac{- y e^{\frac{x}{y}}}{- x e^{\frac{x}{y}}}$

Step 5.3.2.2.2.2

Rewrite the expression.

$\frac{y' e^{\frac{x}{y}}}{e^{\frac{x}{y}}} = \frac{y^{2}}{- x e^{\frac{x}{y}}} + \frac{- y e^{\frac{x}{y}}}{- x e^{\frac{x}{y}}}$

Step 5.3.2.2.3

Cancel the common factor of $e^{\frac{x}{y}}$ .

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Step 5.3.2.2.3.1

Cancel the common factor.

$\frac{y' e^{\frac{x}{y}}}{e^{\frac{x}{y}}} = \frac{y^{2}}{- x e^{\frac{x}{y}}} + \frac{- y e^{\frac{x}{y}}}{- x e^{\frac{x}{y}}}$

Step 5.3.2.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{y^{2}}{- x e^{\frac{x}{y}}} + \frac{- y e^{\frac{x}{y}}}{- x e^{\frac{x}{y}}}$

Step 5.3.2.3

Simplify the right side.

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Step 5.3.2.3.1

Simplify each term.

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Step 5.3.2.3.1.1

Move the negative in front of the fraction.

$y' = - \frac{y^{2}}{x e^{\frac{x}{y}}} + \frac{- y e^{\frac{x}{y}}}{- x e^{\frac{x}{y}}}$

Step 5.3.2.3.1.2

Dividing two negative values results in a positive value.

$y' = - \frac{y^{2}}{x e^{\frac{x}{y}}} + \frac{y e^{\frac{x}{y}}}{x e^{\frac{x}{y}}}$

Step 5.3.2.3.1.3

Cancel the common factor of $e^{\frac{x}{y}}$ .

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Step 5.3.2.3.1.3.1

Cancel the common factor.

$y' = - \frac{y^{2}}{x e^{\frac{x}{y}}} + \frac{y e^{\frac{x}{y}}}{x e^{\frac{x}{y}}}$

Step 5.3.2.3.1.3.2

Rewrite the expression.

$y' = - \frac{y^{2}}{x e^{\frac{x}{y}}} + \frac{y}{x}$

Step 6

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

$\frac{d y}{d x} = - \frac{y^{2}}{x e^{\frac{x}{y}}} + \frac{y}{x}$