Calculus Examples

$e^{x} y = 1$

Step 1

Differentiate both sides of the equation.

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

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

$0$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

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

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Divide $y'$ by $1$ .

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

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

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

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Step 5.3.3.1.1

Cancel the common factor.

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

Step 5.3.3.1.2

Divide $- y$ by $1$ .

$y' = - y$

Step 6

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

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