Calculus Examples

$\frac{d y}{d x} = \frac{6 e^{2 x}}{e^{y}}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $e^{y}$ .

$e^{y} \frac{d y}{d x} = e^{y} \frac{6 e^{2 x}}{e^{y}}$

Step 1.2

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

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Step 1.2.1

Cancel the common factor.

$e^{y} \frac{d y}{d x} = e^{y} \frac{6 e^{2 x}}{e^{y}}$

Step 1.2.2

Rewrite the expression.

$e^{y} \frac{d y}{d x} = 6 e^{2 x}$

Step 1.3

Rewrite the equation.

$e^{y} d y = 6 e^{2 x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int e^{y} d y = \int 6 e^{2 x} d x$

Step 2.2

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

$e^{y} + C_{1} = \int 6 e^{2 x} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$e^{y} + C_{1} = 6 \int e^{2 x} d x$

Step 2.3.2

Let $u = 2 x$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.2.1

Let $u = 2 x$ . Find $\frac{d u}{d x}$ .

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Step 2.3.2.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 2.3.2.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 2.3.2.1.3

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

$2 \cdot 1$

Step 2.3.2.1.4

Multiply $2$ by $1$ .

$2$

Step 2.3.2.2

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

$e^{y} + C_{1} = 6 \int e^{u} \frac{1}{2} d u$

Step 2.3.3

Combine $e^{u}$ and $\frac{1}{2}$ .

$e^{y} + C_{1} = 6 \int \frac{e^{u}}{2} d u$

Step 2.3.4

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$e^{y} + C_{1} = 6 (\frac{1}{2} \int e^{u} d u)$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Combine $\frac{1}{2}$ and $6$ .

$e^{y} + C_{1} = \frac{6}{2} \int e^{u} d u$

Step 2.3.5.2

Cancel the common factor of $6$ and $2$ .

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Step 2.3.5.2.1

Factor $2$ out of $6$ .

$e^{y} + C_{1} = \frac{2 \cdot 3}{2} \int e^{u} d u$

Step 2.3.5.2.2

Cancel the common factors.

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Step 2.3.5.2.2.1

Factor $2$ out of $2$ .

$e^{y} + C_{1} = \frac{2 \cdot 3}{2 (1)} \int e^{u} d u$

Step 2.3.5.2.2.2

Cancel the common factor.

$e^{y} + C_{1} = \frac{2 \cdot 3}{2 \cdot 1} \int e^{u} d u$

Step 2.3.5.2.2.3

Rewrite the expression.

$e^{y} + C_{1} = \frac{3}{1} \int e^{u} d u$

Step 2.3.5.2.2.4

Divide $3$ by $1$ .

$e^{y} + C_{1} = 3 \int e^{u} d u$

Step 2.3.6

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

$e^{y} + C_{1} = 3 (e^{u} + C_{2})$

Step 2.3.7

Simplify.

$e^{y} + C_{1} = 3 e^{u} + C_{2}$

Step 2.3.8

Replace all occurrences of $u$ with $2 x$ .

$e^{y} + C_{1} = 3 e^{2 x} + C_{2}$

Step 2.4

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

$e^{y} = 3 e^{2 x} + K$

Step 3

Solve for $y$ .

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Step 3.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{y}) = \ln (3 e^{2 x} + K)$

Step 3.2

Expand the left side.

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Step 3.2.1

Expand $\ln (e^{y})$ by moving $y$ outside the logarithm.

$y \ln (e) = \ln (3 e^{2 x} + K)$

Step 3.2.2

The natural logarithm of $e$ is $1$ .

$y \cdot 1 = \ln (3 e^{2 x} + K)$

Step 3.2.3

Multiply $y$ by $1$ .

$y = \ln (3 e^{2 x} + K)$