Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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 $g (x) = x - y$ .

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

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

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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Step 3.1.1

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

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x - y]$

Step 3.1.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

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

Step 3.1.3

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

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

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

$\frac{1}{x - y} (1 - y')$

Step 3.4

Simplify.

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Step 3.4.1

Reorder the factors of $\frac{1}{x - y} (1 - y')$ .

$(1 - y') \frac{1}{x - y}$

Step 3.4.2

Multiply $1 - y'$ by $\frac{1}{x - y}$ .

$\frac{1 - y'}{x - y}$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Multiply both sides by $x - y$ .

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

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

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Step 5.2.1.1.1

Simplify by multiplying through.

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Step 5.2.1.1.1.1

Apply the distributive property.

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

Step 5.2.1.1.1.2

Simplify the expression.

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Step 5.2.1.1.1.2.1

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

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

Step 5.2.1.1.1.2.2

Rewrite using the commutative property of multiplication.

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

Step 5.2.1.1.2

Expand $(e^{x - y} - e^{x - y} y') (x - y)$ using the FOIL Method.

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Step 5.2.1.1.2.1

Apply the distributive property.

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

Step 5.2.1.1.2.2

Apply the distributive property.

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

Step 5.2.1.1.2.3

Apply the distributive property.

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

Step 5.2.1.1.3

Simplify terms.

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Step 5.2.1.1.3.1

Simplify each term.

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Step 5.2.1.1.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 5.2.1.1.3.1.2

Multiply $- e^{x - y} y' (- y)$ .

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Step 5.2.1.1.3.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 5.2.1.1.3.1.2.2

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

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

Step 5.2.1.1.3.2

Simplify the expression.

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Step 5.2.1.1.3.2.1

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

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

Step 5.2.1.1.3.2.2

Move $- y e^{x - y}$ .

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

Step 5.2.1.1.3.2.3

Move $x e^{x - y}$ .

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $\frac{1 - y'}{x - y} (x - y)$ .

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Step 5.2.2.1.1

Cancel the common factor of $x - y$ .

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Step 5.2.2.1.1.1

Cancel the common factor.

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

Step 5.2.2.1.1.2

Rewrite the expression.

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

Step 5.2.2.1.2

Reorder $1$ and $- y'$ .

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

Step 5.3

Solve for $y'$ .

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Step 5.3.1

Add $y'$ to both sides of the equation.

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

Step 5.3.2

Move all terms not containing $y'$ to the right side of the equation.

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Step 5.3.2.1

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

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

Step 5.3.2.2

Add $y e^{x - y}$ to both sides of the equation.

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

Step 5.3.3

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

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Step 5.3.3.1

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

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

Step 5.3.3.2

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

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

Step 5.3.3.3

Raise $y'$ to the power of $1$ .

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

Step 5.3.3.4

Factor $y'$ out of ${y'}^{1}$ .

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

Step 5.3.3.5

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

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

Step 5.3.3.6

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

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

Step 5.3.4

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

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

Step 5.3.5

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

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Step 5.3.5.1

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

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

Step 5.3.5.2

Simplify the left side.

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Step 5.3.5.2.1

Cancel the common factor of $- x e^{x - y} + y e^{x - y} + 1$ .

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Step 5.3.5.2.1.1

Cancel the common factor.

Step 5.3.5.2.1.2

Divide $y'$ by $1$ .

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

Step 5.3.5.3

Simplify the right side.

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Step 5.3.5.3.1

Move the negative in front of the fraction.

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

Step 5.3.5.3.2

Combine the numerators over the common denominator.

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

Step 5.3.5.3.3

Combine the numerators over the common denominator.

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

Step 5.3.5.3.4

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

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

Step 5.3.5.3.5

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

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

Step 5.3.5.3.6

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

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

Step 5.3.5.3.7

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 5.3.5.3.8

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

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

Step 5.3.5.3.9

Rewrite negatives.

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Step 5.3.5.3.9.1

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

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

Step 5.3.5.3.9.2

Move the negative in front of the fraction.

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

Step 6

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

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