Calculus Examples

e^{y} = x y

$e^{y} = x y$

Step 1

Differentiate both sides of the equation.

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

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

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

$g (x) = y$ .

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Step 2.1.1

To apply the Chain Rule, set

$u$ as

$y$ .

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

Step 2.1.2

Differentiate using the Exponential Rule which states that

$\frac{d}{d u} [a^{u}]$ is

$a^{u} \ln (a)$ where

$a$ =

$e$ .

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

Step 2.1.3

Replace all occurrences of

$u$ with

$y$ .

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

Step 2.2

Rewrite

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

$y'$ .

$e^{y} y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 3.2

Rewrite

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

$y'$ .

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

Step 3.3

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 3.4

Multiply

$y$ by

$1$ .

$x y' + y$

Step 4

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

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

Step 5

Solve for

$y'$ .

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Step 5.1

Reorder factors in

$e^{y} y'$ .

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

Step 5.2

Subtract

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

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

Step 5.3

Factor

$y'$ out of

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

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Step 5.3.1

Factor

$y'$ out of

$y' e^{y}$ .

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

Step 5.3.2

Factor

$y'$ out of

$- x y'$ .

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

Step 5.3.3

Factor

$y'$ out of

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

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

Step 5.4

Divide each term in

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

$e^{y} - x$ and simplify.

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Step 5.4.1

Divide each term in

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

$e^{y} - x$ .

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

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of

$e^{y} - x$ .

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Step 5.4.2.1.1

Cancel the common factor.

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

Step 5.4.2.1.2

Divide

$y'$ by

$1$ .

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

Step 6

Replace

$y'$ with

$\frac{d y}{d x}$ .

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