Calculus Examples

$x y - e^{x y} = 7$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (x y - e^{x y}) = \frac{d}{d y} (7)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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 $g (y) = y$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $x$ by $1$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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 2.3.2.1

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

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

Step 2.3.2.2

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

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

Step 2.3.2.3

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

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

Step 2.3.3

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 $g (y) = y$ .

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

Step 2.3.4

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

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

Step 2.3.5

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

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

Step 2.3.6

Multiply $x$ by $1$ .

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Reorder terms.

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

Step 3

Since $7$ is constant with respect to $y$ , the derivative of $7$ with respect to $y$ is $0$ .

$0$

Step 4

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

$y x' - e^{x y} x - e^{x y} y x' + x = 0$

Step 5

Solve for $x'$ .

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Step 5.1

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

$y x' - x e^{x y} - y x' e^{x y} + x = 0$

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Subtract $x$ from both sides of the equation.

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

Step 5.3

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

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Step 5.3.1

Factor $y x'$ out of $y x'$ .

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

Step 5.3.2

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

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

Step 5.3.3

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

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

Step 5.4

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

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

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $y$ .

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Step 5.5.2.1.1

Cancel the common factor.

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

Step 5.5.2.1.2

Rewrite the expression.

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

Step 5.5.2.2

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

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Step 5.5.2.2.1

Cancel the common factor.

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

Step 5.5.2.2.2

Divide $x'$ by $1$ .

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

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Move the negative in front of the fraction.

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

Step 6

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

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