Calculus Examples

$y = x e^{y}$

Differentiate both sides of the equation.

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

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Differentiate the right side of the equation.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Solve for $y'$ .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Simplify the left side.

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

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Cancel the common factor.

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

Divide $y'$ by $1$ .

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

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

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