Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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 $g (y) = y$ .

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Multiply $x$ by $1$ .

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Remove unnecessary parentheses.

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

Step 2.4.3

Reorder terms.

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

Step 3

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

$0$

Step 4

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

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

Step 5

Solve for $x'$ .

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Step 5.1

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

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Step 5.1.1

Add $x$ to both sides of the equation.

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

Step 5.1.2

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

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

Step 5.2.2.2

Cancel the common factor of $y$ .

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Step 5.2.2.2.1

Cancel the common factor.

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

Step 5.2.2.2.2

Divide $x'$ by $1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Move the negative in front of the fraction.

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

Step 5.2.3.1.2

Dividing two negative values results in a positive value.

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

Step 6

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

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