Calculus Examples

$f (x) = e^{x y}$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate using the function rule which states that the derivative of $f (x)$ is $\frac{d f}{d x}$ .

$\frac{d f}{d x}$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

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 3.3

Differentiate using the Power Rule.

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Step 3.3.1

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 3.3.2

Multiply $x$ by $1$ .

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

Step 3.4

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

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

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

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

Step 3.5.2

Reorder terms.

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

Step 4

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

$\frac{d f}{d x} = e^{x y} x + e^{x y} y x'$

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $e^{x y} x + e^{x y} y x' = \frac{d f}{d x}$ .

$e^{x y} x + e^{x y} y x' = \frac{d f}{d x}$

Step 5.2

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

$x e^{x y} + y x' e^{x y} = \frac{d f}{d x}$

Step 5.3

Cancel the common factor of $d$ .

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Step 5.3.1

Cancel the common factor.

$x e^{x y} + y x' e^{x y} = \frac{d f}{d x}$

Step 5.3.2

Rewrite the expression.

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

Step 5.4

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

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

Step 5.5

Divide each term in $y x' e^{x y} = \frac{f}{x} - x e^{x y}$ by $y e^{x y}$ and simplify.

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Step 5.5.1

Divide each term in $y x' e^{x y} = \frac{f}{x} - x e^{x y}$ by $y e^{x y}$ .

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

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.

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

Step 5.5.2.1.2

Rewrite the expression.

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

Step 5.5.2.2

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

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Step 5.5.2.2.1

Cancel the common factor.

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

Step 5.5.2.2.2

Divide $x'$ by $1$ .

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

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Simplify each term.

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Step 5.5.3.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.5.3.1.2

Combine.

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

Step 5.5.3.1.3

Multiply $f$ by $1$ .

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

Step 5.5.3.1.4

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

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Step 5.5.3.1.4.1

Cancel the common factor.

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

Step 5.5.3.1.4.2

Rewrite the expression.

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

Step 5.5.3.1.5

Move the negative in front of the fraction.

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

Step 6

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

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