Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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 $g (x) = e^{y}$ .

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

Step 2.2

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 $g (x) = y$ .

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

$y' + 0$

Step 3.4

Add $y'$ and $0$ .

$y'$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

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

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

Step 5.2

Subtract $y'$ from both sides of the equation.

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

Step 5.3

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

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

Factor $y'$ out of $- y'$ .

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

Step 5.4.3

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

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

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

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

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Step 5.5.2.1.1

Cancel the common factor.

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

Step 5.5.2.1.2

Divide $y'$ by $1$ .

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

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.

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

Step 6

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

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