Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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 the right side of the equation.

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Step 3.1

Differentiate.

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Step 3.1.1

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

$\frac{d}{d x} [x] + \frac{d}{d x} [- y]$

Step 3.1.2

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

$1 + \frac{d}{d x} [- y]$

Step 3.2

Evaluate $\frac{d}{d x} [- y]$ .

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Step 3.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- y$ with respect to $x$ is $- \frac{d}{d x} [y]$ .

$1 - \frac{d}{d x} [y]$

Step 3.2.2

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

$1 - y'$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

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

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

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.

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

Step 5.2.1.1.1.2

Rewrite the expression.

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

Step 5.2.1.1.2

Apply the distributive property.

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

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.

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

Step 5.2.1.1.3.2

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

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

Step 5.2.1.1.3.3

Move $x$ .

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

Step 5.2.1.1.3.4

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

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $(1 - y') y^{2}$ .

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Step 5.2.2.1.1

Apply the distributive property.

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

Step 5.2.2.1.2

Simplify the expression.

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Step 5.2.2.1.2.1

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

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

Step 5.2.2.1.2.2

Reorder $y^{2}$ and $- y' y^{2}$ .

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

Step 5.3

Solve for $y'$ .

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Step 5.3.1

Add $y' y^{2}$ to both sides of the equation.

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

Step 5.3.2

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

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

Step 5.3.3

Factor $y'$ out of $- y' x e^{\frac{x}{y}} + y' y^{2}$ .

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Step 5.3.3.1

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

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

Step 5.3.3.2

Factor $y'$ out of $y' y^{2}$ .

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

Step 5.3.3.3

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

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

Step 5.3.4

Rewrite $- 1 x$ as $- x$ .

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

Step 5.3.5

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

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Step 5.3.5.1

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

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

Step 5.3.5.2

Simplify the left side.

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Step 5.3.5.2.1

Cancel the common factor of $- x e^{\frac{x}{y}} + y^{2}$ .

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Step 5.3.5.2.1.1

Cancel the common factor.

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

Step 5.3.5.2.1.2

Divide $y'$ by $1$ .

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

Step 5.3.5.3

Simplify the right side.

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Step 5.3.5.3.1

Combine the numerators over the common denominator.

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

Step 5.3.5.3.2

Simplify the numerator.

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Step 5.3.5.3.2.1

Factor $y$ out of $y^{2} - y e^{\frac{x}{y}}$ .

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Step 5.3.5.3.2.1.1

Factor $y$ out of $y^{2}$ .

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

Step 5.3.5.3.2.1.2

Factor $y$ out of $- y e^{\frac{x}{y}}$ .

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

Step 5.3.5.3.2.1.3

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

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

Step 5.3.5.3.2.2

Rewrite $- 1 e^{\frac{x}{y}}$ as $- e^{\frac{x}{y}}$ .

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

Step 5.3.5.3.3

Simplify with factoring out.

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Step 5.3.5.3.3.1

Factor $- 1$ out of $- x e^{\frac{x}{y}}$ .

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

Step 5.3.5.3.3.2

Factor $- 1$ out of $y^{2}$ .

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

Step 5.3.5.3.3.3

Factor $- 1$ out of $- (x e^{\frac{x}{y}}) - 1 (- y^{2})$ .

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

Step 5.3.5.3.3.4

Rewrite negatives.

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Step 5.3.5.3.3.4.1

Rewrite $- (x e^{\frac{x}{y}} - y^{2})$ as $- 1 (x e^{\frac{x}{y}} - y^{2})$ .

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

Step 5.3.5.3.3.4.2

Move the negative in front of the fraction.

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

Step 6

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

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