Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Regroup factors.

$\frac{d y}{d x} = \frac{4}{x} \cdot \frac{1}{e^{y}}$

Step 1.2

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

$e^{y} \frac{d y}{d x} = e^{y} (\frac{4}{x} \cdot \frac{1}{e^{y}})$

Step 1.3

Simplify.

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Step 1.3.1

Combine.

$e^{y} \frac{d y}{d x} = e^{y} \frac{4 \cdot 1}{x e^{y}}$

Step 1.3.2

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

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Step 1.3.2.1

Factor $e^{y}$ out of $x e^{y}$ .

$e^{y} \frac{d y}{d x} = e^{y} \frac{4 \cdot 1}{e^{y} x}$

Step 1.3.2.2

Cancel the common factor.

$e^{y} \frac{d y}{d x} = e^{y} \frac{4 \cdot 1}{e^{y} x}$

Step 1.3.2.3

Rewrite the expression.

$e^{y} \frac{d y}{d x} = \frac{4 \cdot 1}{x}$

Step 1.3.3

Multiply $4$ by $1$ .

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

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.

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

Step 2.3.2

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$e^{y} + C_{1} = 4 (\ln (| x |) + C_{2})$

Step 2.3.3

Simplify.

$e^{y} + C_{1} = 4 \ln (| x |) + C_{2}$

Step 2.4

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

$e^{y} = 4 \ln (| x |) + C$

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.

$\ln (e^{y}) = \ln (\ln (x^{4}) + C)$

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.

$y \ln (e) = \ln (\ln (x^{4}) + C)$

Step 3.2.2

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

$y \cdot 1 = \ln (\ln (x^{4}) + C)$

Step 3.2.3

Multiply $y$ by $1$ .

$y = \ln (\ln (x^{4}) + C)$

Step 3.3

Expand $\ln (x^{4})$ by moving $4$ outside the logarithm.

$y = \ln (4 \ln (x) + C)$

Step 3.4

Simplify $4 \ln (x)$ by moving $4$ inside the logarithm.

$y = \ln (\ln (x^{4}) + C)$