Calculus Examples

$d x - e^{x} d y = 0$

Step 1

Subtract $d x$ from both sides of the equation.

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

Step 2

Multiply both sides by $\frac{1}{e^{x}}$ .

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

Step 3

Simplify.

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Step 3.1

Rewrite using the commutative property of multiplication.

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

Step 3.2

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

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Step 3.2.1

Move the leading negative in $- \frac{1}{e^{x}}$ into the numerator.

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

Combine $\frac{1}{e^{x}}$ and $- 1$ .

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

Step 3.4

Move the negative in front of the fraction.

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

Apply the constant rule.

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

Step 4.3

Integrate the right side.

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Step 4.3.1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 4.3.2

Simplify the expression.

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Step 4.3.2.1

Negate the exponent of $e^{x}$ and move it out of the denominator.

$- y + C_{1} = - \int 1 {(e^{x})}^{- 1} d x$

Step 4.3.2.2

Simplify.

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Step 4.3.2.2.1

Multiply the exponents in ${(e^{x})}^{- 1}$ .

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Step 4.3.2.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- y + C_{1} = - \int 1 e^{x \cdot - 1} d x$

Step 4.3.2.2.1.2

Move $- 1$ to the left of $x$ .

$- y + C_{1} = - \int 1 e^{- 1 \cdot x} d x$

Step 4.3.2.2.1.3

Rewrite $- 1 x$ as $- x$ .

$- y + C_{1} = - \int 1 e^{- x} d x$

Step 4.3.2.2.2

Multiply $e^{- x}$ by $1$ .

$- y + C_{1} = - \int e^{- x} d x$

Step 4.3.3

Let $u = - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.3.3.1

Let $u = - x$ . Find $\frac{d u}{d x}$ .

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Step 4.3.3.1.1

Differentiate $- x$ .

$\frac{d}{d x} [- x]$

Step 4.3.3.1.2

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

$- \frac{d}{d x} [x]$

Step 4.3.3.1.3

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

$- 1 \cdot 1$

Step 4.3.3.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 4.3.3.2

Rewrite the problem using $u$ and $d u$ .

$- y + C_{1} = - \int - e^{u} d u$

Step 4.3.4

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- y + C_{1} = - - \int e^{u} d u$

Step 4.3.5

Simplify.

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Step 4.3.5.1

Multiply $- 1$ by $- 1$ .

$- y + C_{1} = 1 \int e^{u} d u$

Step 4.3.5.2

Multiply $\int e^{u} d u$ by $1$ .

$- y + C_{1} = \int e^{u} d u$

Step 4.3.6

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$- y + C_{1} = e^{u} + C_{2}$

Step 4.3.7

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

$- y + C_{1} = e^{- x} + C_{2}$

Step 4.4

Group the constant of integration on the right side as $K$ .

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

Step 5

Divide each term in $- y = e^{- x} + K$ by $- 1$ and simplify.

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Step 5.1

Divide each term in $- y = e^{- x} + K$ by $- 1$ .

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

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

Step 5.2.2

Divide $y$ by $1$ .

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

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Move the negative one from the denominator of $\frac{e^{- x}}{- 1}$ .

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

Step 5.3.1.2

Rewrite $- 1 \cdot e^{- x}$ as $- e^{- x}$ .

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

Step 5.3.1.3

Move the negative one from the denominator of $\frac{K}{- 1}$ .

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

Step 5.3.1.4

Rewrite $- 1 \cdot K$ as $- K$ .

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

Step 6

Simplify the constant of integration.

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