Calculus Examples

$x^{7} + 14 y \frac{d y}{d x} = 0$

Step 1

Separate the variables.

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Step 1.1

Solve for $\frac{d y}{d x}$ .

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Step 1.1.1

Subtract $x^{7}$ from both sides of the equation.

$14 y \frac{d y}{d x} = - x^{7}$

Step 1.1.2

Divide each term in $14 y \frac{d y}{d x} = - x^{7}$ by $14 y$ and simplify.

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Step 1.1.2.1

Divide each term in $14 y \frac{d y}{d x} = - x^{7}$ by $14 y$ .

$\frac{14 y \frac{d y}{d x}}{14 y} = \frac{- x^{7}}{14 y}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $14$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{14 y \frac{d y}{d x}}{14 y} = \frac{- x^{7}}{14 y}$

Step 1.1.2.2.1.2

Rewrite the expression.

$\frac{y \frac{d y}{d x}}{y} = \frac{- x^{7}}{14 y}$

Step 1.1.2.2.2

Cancel the common factor of $y$ .

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Step 1.1.2.2.2.1

Cancel the common factor.

$\frac{y \frac{d y}{d x}}{y} = \frac{- x^{7}}{14 y}$

Step 1.1.2.2.2.2

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

$\frac{d y}{d x} = \frac{- x^{7}}{14 y}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Move the negative in front of the fraction.

$\frac{d y}{d x} = - \frac{x^{7}}{14 y}$

Step 1.2

Multiply both sides by $y$ .

$y \frac{d y}{d x} = y (- \frac{x^{7}}{14 y})$

Step 1.3

Simplify.

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Step 1.3.1

Rewrite using the commutative property of multiplication.

$y \frac{d y}{d x} = - y \frac{x^{7}}{14 y}$

Step 1.3.2

Cancel the common factor of $y$ .

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Step 1.3.2.1

Factor $y$ out of $- y$ .

$y \frac{d y}{d x} = y \cdot - 1 \frac{x^{7}}{14 y}$

Step 1.3.2.2

Factor $y$ out of $14 y$ .

$y \frac{d y}{d x} = y \cdot - 1 \frac{x^{7}}{y \cdot 14}$

Step 1.3.2.3

Cancel the common factor.

$y \frac{d y}{d x} = y \cdot - 1 \frac{x^{7}}{y \cdot 14}$

Step 1.3.2.4

Rewrite the expression.

$y \frac{d y}{d x} = - \frac{x^{7}}{14}$

Step 1.4

Rewrite the equation.

$y d y = - \frac{x^{7}}{14} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y d y = \int - \frac{x^{7}}{14} d x$

Step 2.2

By the Power Rule, the integral of $y$ with respect to $y$ is $\frac{1}{2} y^{2}$ .

$\frac{1}{2} y^{2} + C_{1} = \int - \frac{x^{7}}{14} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

$\frac{1}{2} y^{2} + C_{1} = - \int \frac{x^{7}}{14} d x$

Step 2.3.2

Since $\frac{1}{14}$ is constant with respect to $x$ , move $\frac{1}{14}$ out of the integral.

$\frac{1}{2} y^{2} + C_{1} = - (\frac{1}{14} \int x^{7} d x)$

Step 2.3.3

By the Power Rule, the integral of $x^{7}$ with respect to $x$ is $\frac{1}{8} x^{8}$ .

$\frac{1}{2} y^{2} + C_{1} = - \frac{1}{14} (\frac{1}{8} x^{8} + C_{2})$

Step 2.3.4

Simplify the answer.

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Step 2.3.4.1

Rewrite $- \frac{1}{14} (\frac{1}{8} x^{8} + C_{2})$ as $- \frac{1}{14} \cdot \frac{1}{8} x^{8} + C_{2}$ .

$\frac{1}{2} y^{2} + C_{1} = - \frac{1}{14} \cdot \frac{1}{8} x^{8} + C_{2}$

Step 2.3.4.2

Simplify.

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Step 2.3.4.2.1

Multiply $\frac{1}{8}$ by $\frac{1}{14}$ .

$\frac{1}{2} y^{2} + C_{1} = - \frac{1}{8 \cdot 14} x^{8} + C_{2}$

Step 2.3.4.2.2

Multiply $8$ by $14$ .

$\frac{1}{2} y^{2} + C_{1} = - \frac{1}{112} x^{8} + C_{2}$

Step 2.4

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

$\frac{1}{2} y^{2} = - \frac{1}{112} x^{8} + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} y^{2}) = 2 (- \frac{1}{112} x^{8} + K)$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $2 (\frac{1}{2} y^{2})$ .

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Step 3.2.1.1.1

Combine $\frac{1}{2}$ and $y^{2}$ .

$2 \frac{y^{2}}{2} = 2 (- \frac{1}{112} x^{8} + K)$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$2 \frac{y^{2}}{2} = 2 (- \frac{1}{112} x^{8} + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (- \frac{1}{112} x^{8} + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (- \frac{1}{112} x^{8} + K)$ .

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Step 3.2.2.1.1

Combine $x^{8}$ and $\frac{1}{112}$ .

$y^{2} = 2 (- \frac{x^{8}}{112} + K)$

Step 3.2.2.1.2

Apply the distributive property.

$y^{2} = 2 (- \frac{x^{8}}{112}) + 2 K$

Step 3.2.2.1.3

Cancel the common factor of $2$ .

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Step 3.2.2.1.3.1

Move the leading negative in $- \frac{x^{8}}{112}$ into the numerator.

$y^{2} = 2 \frac{- x^{8}}{112} + 2 K$

Step 3.2.2.1.3.2

Factor $2$ out of $112$ .

$y^{2} = 2 \frac{- x^{8}}{2 (56)} + 2 K$

Step 3.2.2.1.3.3

Cancel the common factor.

$y^{2} = 2 \frac{- x^{8}}{2 \cdot 56} + 2 K$

Step 3.2.2.1.3.4

Rewrite the expression.

$y^{2} = \frac{- x^{8}}{56} + 2 K$

Step 3.2.2.1.4

Move the negative in front of the fraction.

$y^{2} = - \frac{x^{8}}{56} + 2 K$

Step 3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{- \frac{x^{8}}{56} + 2 K}$

Step 3.4

Simplify $\pm \sqrt{- \frac{x^{8}}{56} + 2 K}$ .

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Step 3.4.1

To write $2 K$ as a fraction with a common denominator, multiply by $\frac{56}{56}$ .

$y = \pm \sqrt{- \frac{x^{8}}{56} + 2 K \cdot \frac{56}{56}}$

Step 3.4.2

Combine $2 K$ and $\frac{56}{56}$ .

$y = \pm \sqrt{- \frac{x^{8}}{56} + \frac{2 K \cdot 56}{56}}$

Step 3.4.3

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{- x^{8} + 2 K \cdot 56}{56}}$

Step 3.4.4

Multiply $56$ by $2$ .

$y = \pm \sqrt{\frac{- x^{8} + 112 K}{56}}$

Step 3.4.5

Rewrite $\frac{- x^{8} + 112 K}{56}$ as ${(\frac{1}{2})}^{2} \frac{- x^{8} + 112 K}{14}$ .

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Step 3.4.5.1

Factor the perfect power $1^{2}$ out of $- x^{8} + 112 K$ .

$y = \pm \sqrt{\frac{1^{2} (- x^{8} + 112 K)}{56}}$

Step 3.4.5.2

Factor the perfect power $2^{2}$ out of $56$ .

$y = \pm \sqrt{\frac{1^{2} (- x^{8} + 112 K)}{2^{2} \cdot 14}}$

Step 3.4.5.3

Rearrange the fraction $\frac{1^{2} (- x^{8} + 112 K)}{2^{2} \cdot 14}$ .

$y = \pm \sqrt{{(\frac{1}{2})}^{2} \frac{- x^{8} + 112 K}{14}}$

Step 3.4.6

Pull terms out from under the radical.

$y = \pm \frac{1}{2} \sqrt{\frac{- x^{8} + 112 K}{14}}$

Step 3.4.7

Rewrite $\sqrt{\frac{- x^{8} + 112 K}{14}}$ as $\frac{\sqrt{- x^{8} + 112 K}}{\sqrt{14}}$ .

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{- x^{8} + 112 K}}{\sqrt{14}}$

Step 3.4.8

Combine.

$y = \pm \frac{1 \sqrt{- x^{8} + 112 K}}{2 \sqrt{14}}$

Step 3.4.9

Multiply $\sqrt{- x^{8} + 112 K}$ by $1$ .

$y = \pm \frac{\sqrt{- x^{8} + 112 K}}{2 \sqrt{14}}$

Step 3.4.10

Multiply $\frac{\sqrt{- x^{8} + 112 K}}{2 \sqrt{14}}$ by $\frac{\sqrt{14}}{\sqrt{14}}$ .

$y = \pm \frac{\sqrt{- x^{8} + 112 K}}{2 \sqrt{14}} \cdot \frac{\sqrt{14}}{\sqrt{14}}$

Step 3.4.11

Combine and simplify the denominator.

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Step 3.4.11.1

Multiply $\frac{\sqrt{- x^{8} + 112 K}}{2 \sqrt{14}}$ by $\frac{\sqrt{14}}{\sqrt{14}}$ .

$y = \pm \frac{\sqrt{- x^{8} + 112 K} \sqrt{14}}{2 \sqrt{14} \sqrt{14}}$

Step 3.4.11.2

Move $\sqrt{14}$ .

$y = \pm \frac{\sqrt{- x^{8} + 112 K} \sqrt{14}}{2 (\sqrt{14} \sqrt{14})}$

Step 3.4.11.3

Raise $\sqrt{14}$ to the power of $1$ .

$y = \pm \frac{\sqrt{- x^{8} + 112 K} \sqrt{14}}{2 ({\sqrt{14}}^{1} \sqrt{14})}$

Step 3.4.11.4

Raise $\sqrt{14}$ to the power of $1$ .

$y = \pm \frac{\sqrt{- x^{8} + 112 K} \sqrt{14}}{2 ({\sqrt{14}}^{1} {\sqrt{14}}^{1})}$

Step 3.4.11.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm \frac{\sqrt{- x^{8} + 112 K} \sqrt{14}}{2 {\sqrt{14}}^{1 + 1}}$

Step 3.4.11.6

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{- x^{8} + 112 K} \sqrt{14}}{2 {\sqrt{14}}^{2}}$

Step 3.4.11.7

Rewrite ${\sqrt{14}}^{2}$ as $14$ .

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Step 3.4.11.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{14}$ as $14^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{- x^{8} + 112 K} \sqrt{14}}{2 {(14^{\frac{1}{2}})}^{2}}$

Step 3.4.11.7.2

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

$y = \pm \frac{\sqrt{- x^{8} + 112 K} \sqrt{14}}{2 \cdot 14^{\frac{1}{2} \cdot 2}}$

Step 3.4.11.7.3

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

$y = \pm \frac{\sqrt{- x^{8} + 112 K} \sqrt{14}}{2 \cdot 14^{\frac{2}{2}}}$

Step 3.4.11.7.4

Cancel the common factor of $2$ .

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Step 3.4.11.7.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{- x^{8} + 112 K} \sqrt{14}}{2 \cdot 14^{\frac{2}{2}}}$

Step 3.4.11.7.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{- x^{8} + 112 K} \sqrt{14}}{2 \cdot 14^{1}}$

Step 3.4.11.7.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{- x^{8} + 112 K} \sqrt{14}}{2 \cdot 14}$

Step 3.4.12

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{(- x^{8} + 112 K) \cdot 14}}{2 \cdot 14}$

Step 3.4.13

Simplify the expression.

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Step 3.4.13.1

Multiply $2$ by $14$ .

$y = \pm \frac{\sqrt{(- x^{8} + 112 K) \cdot 14}}{28}$

Step 3.4.13.2

Reorder factors in $\pm \frac{\sqrt{(- x^{8} + 112 K) \cdot 14}}{28}$ .

$y = \pm \frac{\sqrt{14 (- x^{8} + 112 K)}}{28}$

Step 3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \frac{\sqrt{14 (- x^{8} + 112 K)}}{28}$

Step 3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \frac{\sqrt{14 (- x^{8} + 112 K)}}{28}$

Step 3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \frac{\sqrt{14 (- x^{8} + 112 K)}}{28}$

$y = - \frac{\sqrt{14 (- x^{8} + 112 K)}}{28}$

$y = \frac{\sqrt{14 (- x^{8} + 112 K)}}{28}$

$y = - \frac{\sqrt{14 (- x^{8} + 112 K)}}{28}$

$y = \frac{\sqrt{14 (- x^{8} + 112 K)}}{28}$

$y = - \frac{\sqrt{14 (- x^{8} + 112 K)}}{28}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt{14 (- x^{8} + K)}}{28}$

$y = - \frac{\sqrt{14 (- x^{8} + K)}}{28}$