Calculus Examples

$\frac{d y}{d x} = \frac{6 x^{5}}{3 e^{y}}$

Step 1

Separate the variables.

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Step 1.1

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

$e^{y} \frac{d y}{d x} = e^{y} \frac{6 x^{5}}{3 e^{y}}$

Step 1.2

Simplify.

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Step 1.2.1

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

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Step 1.2.1.1

Factor $e^{y}$ out of $3 e^{y}$ .

$e^{y} \frac{d y}{d x} = e^{y} \frac{6 x^{5}}{e^{y} \cdot 3}$

Step 1.2.1.2

Cancel the common factor.

$e^{y} \frac{d y}{d x} = e^{y} \frac{6 x^{5}}{e^{y} \cdot 3}$

Step 1.2.1.3

Rewrite the expression.

$e^{y} \frac{d y}{d x} = \frac{6 x^{5}}{3}$

Step 1.2.2

Cancel the common factor of $6$ and $3$ .

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Step 1.2.2.1

Factor $3$ out of $6 x^{5}$ .

$e^{y} \frac{d y}{d x} = \frac{3 (2 x^{5})}{3}$

Step 1.2.2.2

Cancel the common factors.

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Step 1.2.2.2.1

Factor $3$ out of $3$ .

$e^{y} \frac{d y}{d x} = \frac{3 (2 x^{5})}{3 (1)}$

Step 1.2.2.2.2

Cancel the common factor.

$e^{y} \frac{d y}{d x} = \frac{3 (2 x^{5})}{3 \cdot 1}$

Step 1.2.2.2.3

Rewrite the expression.

$e^{y} \frac{d y}{d x} = \frac{2 x^{5}}{1}$

Step 1.2.2.2.4

Divide $2 x^{5}$ by $1$ .

$e^{y} \frac{d y}{d x} = 2 x^{5}$

Step 1.3

Rewrite the equation.

$e^{y} d y = 2 x^{5} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int e^{y} d y = \int 2 x^{5} d x$

Step 2.2

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

$e^{y} + C_{1} = \int 2 x^{5} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$e^{y} + C_{1} = 2 \int x^{5} d x$

Step 2.3.2

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

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

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $2 (\frac{1}{6} x^{6} + C_{2})$ as $2 (\frac{1}{6}) x^{6} + C_{2}$ .

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

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

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

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

Step 2.3.3.2.2

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

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Step 2.3.3.2.2.1

Factor $2$ out of $2$ .

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

Step 2.3.3.2.2.2

Cancel the common factors.

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Step 2.3.3.2.2.2.1

Factor $2$ out of $6$ .

$e^{y} + C_{1} = \frac{2 \cdot 1}{2 \cdot 3} x^{6} + C_{2}$

Step 2.3.3.2.2.2.2

Cancel the common factor.

$e^{y} + C_{1} = \frac{2 \cdot 1}{2 \cdot 3} x^{6} + C_{2}$

Step 2.3.3.2.2.2.3

Rewrite the expression.

$e^{y} + C_{1} = \frac{1}{3} x^{6} + C_{2}$

Step 2.4

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

$e^{y} = \frac{1}{3} x^{6} + K$

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.

$\ln (e^{y}) = \ln (\frac{x^{6}}{3} + K)$

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.

$y \ln (e) = \ln (\frac{x^{6}}{3} + K)$

Step 3.2.2

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

$y \cdot 1 = \ln (\frac{x^{6}}{3} + K)$

Step 3.2.3

Multiply $y$ by $1$ .

$y = \ln (\frac{x^{6}}{3} + K)$