Calculus Examples

$e^{x} \frac{d y}{d x} = 4$

Step 1

Separate the variables.

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Step 1.1

Divide each term in $e^{x} \frac{d y}{d x} = 4$ by $e^{x}$ and simplify.

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Step 1.1.1

Divide each term in $e^{x} \frac{d y}{d x} = 4$ by $e^{x}$ .

$\frac{e^{x} \frac{d y}{d x}}{e^{x}} = \frac{4}{e^{x}}$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

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

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{e^{x} \frac{d y}{d x}}{e^{x}} = \frac{4}{e^{x}}$

Step 1.1.2.1.2

Divide $\frac{d y}{d x}$ by $1$ .

$\frac{d y}{d x} = \frac{4}{e^{x}}$

Step 1.2

Rewrite the equation.

$d y = \frac{4}{e^{x}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{4}{e^{x}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \frac{4}{e^{x}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$y + C_{1} = 4 \int \frac{1}{e^{x}} d x$

Step 2.3.2

Simplify the expression.

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Step 2.3.2.1

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

$y + C_{1} = 4 \int 1 {(e^{x})}^{- 1} d x$

Step 2.3.2.2

Simplify.

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Step 2.3.2.2.1

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

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Step 2.3.2.2.1.1

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

$y + C_{1} = 4 \int 1 e^{x \cdot - 1} d x$

Step 2.3.2.2.1.2

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

$y + C_{1} = 4 \int 1 e^{- 1 \cdot x} d x$

Step 2.3.2.2.1.3

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

$y + C_{1} = 4 \int 1 e^{- x} d x$

Step 2.3.2.2.2

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

$y + C_{1} = 4 \int e^{- x} d x$

Step 2.3.3

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

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Step 2.3.3.1

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

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Step 2.3.3.1.1

Differentiate $- x$ .

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

Step 2.3.3.1.2

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 2.3.3.1.3

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 2.3.3.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 2.3.3.2

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

$y + C_{1} = 4 \int - e^{u} d u$

Step 2.3.4

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

$y + C_{1} = 4 (- \int e^{u} d u)$

Step 2.3.5

Multiply $- 1$ by $4$ .

$y + C_{1} = - 4 \int e^{u} d u$

Step 2.3.6

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

$y + C_{1} = - 4 (e^{u} + C_{2})$

Step 2.3.7

Simplify.

$y + C_{1} = - 4 e^{u} + C_{2}$

Step 2.3.8

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

$y + C_{1} = - 4 e^{- x} + C_{2}$

Step 2.4

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

$y = - 4 e^{- x} + K$