Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

$x y' + y \cdot 1$

Step 2.4

Multiply $y$ by $1$ .

$x y' + y$

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 $f (x) = e^{x}$ and $g (x) = x y - 1$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $x y - 1$ .

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

Step 3.1.2

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

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

Step 3.1.3

Replace all occurrences of $u$ with $x y - 1$ .

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

Step 3.2

By the Sum Rule, the derivative of $x y - 1$ with respect to $x$ is $\frac{d}{d x} [x y] + \frac{d}{d x} [- 1]$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Multiply $y$ by $1$ .

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

Step 3.7

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$e^{x y - 1} (x y' + y + 0)$

Step 3.8

Add $x y' + y$ and $0$ .

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

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

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

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Step 5.1.1

Rewrite.

$x y' + y = 0 + 0 + e^{x y - 1} (x y' + y)$

Step 5.1.2

Simplify by adding zeros.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.1.4

Reorder factors in $e^{x y - 1} (x y') + e^{x y - 1} y$ .

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

Step 5.2

Subtract $x y' e^{x y - 1}$ from both sides of the equation.

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

Step 5.3

Subtract $y$ from both sides of the equation.

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

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

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

Step 5.4.3

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

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

Step 5.5

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

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

Step 5.6

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

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Step 5.6.1

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

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

Step 5.6.2

Simplify the left side.

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Step 5.6.2.1

Cancel the common factor of $x$ .

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Step 5.6.2.1.1

Cancel the common factor.

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

Step 5.6.2.1.2

Rewrite the expression.

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

Step 5.6.2.2

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

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Step 5.6.2.2.1

Cancel the common factor.

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

Step 5.6.2.2.2

Divide $y'$ by $1$ .

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

Step 5.6.3

Simplify the right side.

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Step 5.6.3.1

Combine the numerators over the common denominator.

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

Step 5.6.3.2

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

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Step 5.6.3.2.1

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

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

Step 5.6.3.2.2

Factor $y$ out of $- y$ .

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

Step 5.6.3.2.3

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

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

Step 5.6.3.3

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

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Step 5.6.3.3.1

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

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

Step 5.6.3.3.2

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

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

Step 5.6.3.3.3

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

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

Step 5.6.3.3.4

Reorder terms.

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

Step 5.6.3.3.5

Cancel the common factor.

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

Step 5.6.3.3.6

Rewrite the expression.

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

Step 5.6.3.4

Simplify the expression.

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Step 5.6.3.4.1

Move $- 1$ to the left of $y$ .

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

Step 5.6.3.4.2

Move the negative in front of the fraction.

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

Step 6

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

$\frac{d y}{d x} = - \frac{y}{x}$