Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.2

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = x$ and $g (y) = y$ .

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

Step 2.3

Differentiate using the Power Rule.

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Step 2.3.1

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

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

Step 2.3.2

Multiply $x$ by $1$ .

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

Step 2.4

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

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

Step 2.5

Simplify.

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Step 2.5.1

Apply the distributive property.

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

Step 2.5.2

Reorder terms.

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

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 $y - x$ with respect to $y$ is $\frac{d}{d y} [y] + \frac{d}{d y} [- x]$ .

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

Step 3.1.2

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

$1 - x'$

Step 4

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

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

Step 5

Solve for $x'$ .

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

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

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

Step 5.4.3

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

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

Step 5.4.4

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

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

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

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

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Step 5.5.2.1.1

Cancel the common factor.

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

Step 5.5.2.1.2

Divide $x'$ by $1$ .

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

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Combine the numerators over the common denominator.

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

Step 6

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

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