Calculus Examples

$\frac{d y}{d x} = 6 x y^{8}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{y^{8}}$ .

$\frac{1}{y^{8}} \frac{d y}{d x} = \frac{1}{y^{8}} (6 x y^{8})$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

$\frac{1}{y^{8}} \frac{d y}{d x} = 6 \frac{1}{y^{8}} (x y^{8})$

Step 1.2.2

Combine $6$ and $\frac{1}{y^{8}}$ .

$\frac{1}{y^{8}} \frac{d y}{d x} = \frac{6}{y^{8}} (x y^{8})$

Step 1.2.3

Cancel the common factor of $y^{8}$ .

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Step 1.2.3.1

Factor $y^{8}$ out of $x y^{8}$ .

$\frac{1}{y^{8}} \frac{d y}{d x} = \frac{6}{y^{8}} (y^{8} x)$

Step 1.2.3.2

Cancel the common factor.

$\frac{1}{y^{8}} \frac{d y}{d x} = \frac{6}{y^{8}} (y^{8} x)$

Step 1.2.3.3

Rewrite the expression.

$\frac{1}{y^{8}} \frac{d y}{d x} = 6 x$

Step 1.3

Rewrite the equation.

$\frac{1}{y^{8}} d y = 6 x d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y^{8}} d y = \int 6 x d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Apply basic rules of exponents.

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Step 2.2.1.1

Move $y^{8}$ out of the denominator by raising it to the $- 1$ power.

$\int {(y^{8})}^{- 1} d y = \int 6 x d x$

Step 2.2.1.2

Multiply the exponents in ${(y^{8})}^{- 1}$ .

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Step 2.2.1.2.1

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

$\int y^{8 \cdot - 1} d y = \int 6 x d x$

Step 2.2.1.2.2

Multiply $8$ by $- 1$ .

$\int y^{- 8} d y = \int 6 x d x$

Step 2.2.2

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

$- \frac{1}{7} y^{- 7} + C_{1} = \int 6 x d x$

Step 2.2.3

Simplify the answer.

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Step 2.2.3.1

Rewrite $- \frac{1}{7} y^{- 7} + C_{1}$ as $- \frac{1}{7} \cdot \frac{1}{y^{7}} + C_{1}$ .

$- \frac{1}{7} \cdot \frac{1}{y^{7}} + C_{1} = \int 6 x d x$

Step 2.2.3.2

Simplify.

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Step 2.2.3.2.1

Multiply $\frac{1}{y^{7}}$ by $\frac{1}{7}$ .

$- \frac{1}{y^{7} \cdot 7} + C_{1} = \int 6 x d x$

Step 2.2.3.2.2

Move $7$ to the left of $y^{7}$ .

$- \frac{1}{7 y^{7}} + C_{1} = \int 6 x d x$

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.

$- \frac{1}{7 y^{7}} + C_{1} = 6 \int x d x$

Step 2.3.2

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

$- \frac{1}{7 y^{7}} + C_{1} = 6 (\frac{1}{2} x^{2} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

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

$- \frac{1}{7 y^{7}} + C_{1} = 6 (\frac{1}{2}) x^{2} + C_{2}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

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

$- \frac{1}{7 y^{7}} + C_{1} = \frac{6}{2} x^{2} + C_{2}$

Step 2.3.3.2.2

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

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Step 2.3.3.2.2.1

Factor $2$ out of $6$ .

$- \frac{1}{7 y^{7}} + C_{1} = \frac{2 \cdot 3}{2} x^{2} + C_{2}$

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 $2$ .

$- \frac{1}{7 y^{7}} + C_{1} = \frac{2 \cdot 3}{2 (1)} x^{2} + C_{2}$

Step 2.3.3.2.2.2.2

Cancel the common factor.

$- \frac{1}{7 y^{7}} + C_{1} = \frac{2 \cdot 3}{2 \cdot 1} x^{2} + C_{2}$

Step 2.3.3.2.2.2.3

Rewrite the expression.

$- \frac{1}{7 y^{7}} + C_{1} = \frac{3}{1} x^{2} + C_{2}$

Step 2.3.3.2.2.2.4

Divide $3$ by $1$ .

$- \frac{1}{7 y^{7}} + C_{1} = 3 x^{2} + C_{2}$

Step 2.4

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

$- \frac{1}{7 y^{7}} = 3 x^{2} + K$

Step 3

Solve for $y$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$7 y^{7}, 1, 1$

Step 3.1.2

The LCM of one and any expression is the expression.

$7 y^{7}$

Step 3.2

Multiply each term in $- \frac{1}{7 y^{7}} = 3 x^{2} + K$ by $7 y^{7}$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $- \frac{1}{7 y^{7}} = 3 x^{2} + K$ by $7 y^{7}$ .

$- \frac{1}{7 y^{7}} (7 y^{7}) = 3 x^{2} (7 y^{7}) + K (7 y^{7})$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $7 y^{7}$ .

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Step 3.2.2.1.1

Move the leading negative in $- \frac{1}{7 y^{7}}$ into the numerator.

$\frac{- 1}{7 y^{7}} (7 y^{7}) = 3 x^{2} (7 y^{7}) + K (7 y^{7})$

Step 3.2.2.1.2

Cancel the common factor.

$\frac{- 1}{7 y^{7}} (7 y^{7}) = 3 x^{2} (7 y^{7}) + K (7 y^{7})$

Step 3.2.2.1.3

Rewrite the expression.

$- 1 = 3 x^{2} (7 y^{7}) + K (7 y^{7})$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Rewrite using the commutative property of multiplication.

$- 1 = 3 \cdot 7 x^{2} y^{7} + K (7 y^{7})$

Step 3.2.3.1.2

Multiply $3$ by $7$ .

$- 1 = 21 x^{2} y^{7} + K (7 y^{7})$

Step 3.2.3.1.3

Rewrite using the commutative property of multiplication.

$- 1 = 21 x^{2} y^{7} + 7 K y^{7}$

Step 3.3

Solve the equation.

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Step 3.3.1

Rewrite the equation as $21 x^{2} y^{7} + 7 K y^{7} = - 1$ .

$21 x^{2} y^{7} + 7 K y^{7} = - 1$

Step 3.3.2

Factor $7 y^{7}$ out of $21 x^{2} y^{7} + 7 K y^{7}$ .

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Step 3.3.2.1

Factor $7 y^{7}$ out of $21 x^{2} y^{7}$ .

$7 y^{7} (3 x^{2}) + 7 K y^{7} = - 1$

Step 3.3.2.2

Factor $7 y^{7}$ out of $7 K y^{7}$ .

$7 y^{7} (3 x^{2}) + 7 y^{7} K = - 1$

Step 3.3.2.3

Factor $7 y^{7}$ out of $7 y^{7} (3 x^{2}) + 7 y^{7} K$ .

$7 y^{7} (3 x^{2} + K) = - 1$

Step 3.3.3

Divide each term in $7 y^{7} (3 x^{2} + K) = - 1$ by $7 (3 x^{2} + K)$ and simplify.

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Step 3.3.3.1

Divide each term in $7 y^{7} (3 x^{2} + K) = - 1$ by $7 (3 x^{2} + K)$ .

$\frac{7 y^{7} (3 x^{2} + K)}{7 (3 x^{2} + K)} = \frac{- 1}{7 (3 x^{2} + K)}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $7$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{7 y^{7} (3 x^{2} + K)}{7 (3 x^{2} + K)} = \frac{- 1}{7 (3 x^{2} + K)}$

Step 3.3.3.2.1.2

Rewrite the expression.

$\frac{y^{7} (3 x^{2} + K)}{3 x^{2} + K} = \frac{- 1}{7 (3 x^{2} + K)}$

Step 3.3.3.2.2

Cancel the common factor of $3 x^{2} + K$ .

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Step 3.3.3.2.2.1

Cancel the common factor.

$\frac{y^{7} (3 x^{2} + K)}{3 x^{2} + K} = \frac{- 1}{7 (3 x^{2} + K)}$

Step 3.3.3.2.2.2

Divide $y^{7}$ by $1$ .

$y^{7} = \frac{- 1}{7 (3 x^{2} + K)}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Move the negative in front of the fraction.

$y^{7} = - \frac{1}{7 (3 x^{2} + K)}$

Step 3.3.4

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

$y = \sqrt[7]{- \frac{1}{7 (3 x^{2} + K)}}$

Step 3.3.5

Simplify $\sqrt[7]{- \frac{1}{7 (3 x^{2} + K)}}$ .

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Step 3.3.5.1

Rewrite $- \frac{1}{7 (3 x^{2} + K)}$ as ${({(- 1)}^{7})}^{7} \frac{1}{7 (3 x^{2} + K)}$ .

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Step 3.3.5.1.1

Rewrite $- 1$ as ${(- 1)}^{7}$ .

$y = \sqrt[7]{{(- 1)}^{7} \frac{1}{7 (3 x^{2} + K)}}$

Step 3.3.5.1.2

Rewrite $- 1$ as ${(- 1)}^{7}$ .

$y = \sqrt[7]{{({(- 1)}^{7})}^{7} \frac{1}{7 (3 x^{2} + K)}}$

Step 3.3.5.2

Pull terms out from under the radical.

$y = {(- 1)}^{7} \sqrt[7]{\frac{1}{7 (3 x^{2} + K)}}$

Step 3.3.5.3

Raise $- 1$ to the power of $7$ .

$y = - \sqrt[7]{\frac{1}{7 (3 x^{2} + K)}}$

Step 3.3.5.4

Rewrite $\sqrt[7]{\frac{1}{7 (3 x^{2} + K)}}$ as $\frac{\sqrt[7]{1}}{\sqrt[7]{7 (3 x^{2} + K)}}$ .

$y = - \frac{\sqrt[7]{1}}{\sqrt[7]{7 (3 x^{2} + K)}}$

Step 3.3.5.5

Any root of $1$ is $1$ .

$y = - \frac{1}{\sqrt[7]{7 (3 x^{2} + K)}}$

Step 3.3.5.6

Multiply $\frac{1}{\sqrt[7]{7 (3 x^{2} + K)}}$ by $\frac{{\sqrt[7]{7 (3 x^{2} + K)}}^{6}}{{\sqrt[7]{7 (3 x^{2} + K)}}^{6}}$ .

$y = - (\frac{1}{\sqrt[7]{7 (3 x^{2} + K)}} \cdot \frac{{\sqrt[7]{7 (3 x^{2} + K)}}^{6}}{{\sqrt[7]{7 (3 x^{2} + K)}}^{6}})$

Step 3.3.5.7

Combine and simplify the denominator.

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Step 3.3.5.7.1

Multiply $\frac{1}{\sqrt[7]{7 (3 x^{2} + K)}}$ by $\frac{{\sqrt[7]{7 (3 x^{2} + K)}}^{6}}{{\sqrt[7]{7 (3 x^{2} + K)}}^{6}}$ .

$y = - \frac{{\sqrt[7]{7 (3 x^{2} + K)}}^{6}}{\sqrt[7]{7 (3 x^{2} + K)} {\sqrt[7]{7 (3 x^{2} + K)}}^{6}}$

Step 3.3.5.7.2

Raise $\sqrt[7]{7 (3 x^{2} + K)}$ to the power of $1$ .

$y = - \frac{{\sqrt[7]{7 (3 x^{2} + K)}}^{6}}{{\sqrt[7]{7 (3 x^{2} + K)}}^{1} {\sqrt[7]{7 (3 x^{2} + K)}}^{6}}$

Step 3.3.5.7.3

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

$y = - \frac{{\sqrt[7]{7 (3 x^{2} + K)}}^{6}}{{\sqrt[7]{7 (3 x^{2} + K)}}^{1 + 6}}$

Step 3.3.5.7.4

Add $1$ and $6$ .

$y = - \frac{{\sqrt[7]{7 (3 x^{2} + K)}}^{6}}{{\sqrt[7]{7 (3 x^{2} + K)}}^{7}}$

Step 3.3.5.7.5

Rewrite ${\sqrt[7]{7 (3 x^{2} + K)}}^{7}$ as $7 (3 x^{2} + K)$ .

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Step 3.3.5.7.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{7 (3 x^{2} + K)}$ as ${(7 (3 x^{2} + K))}^{\frac{1}{7}}$ .

$y = - \frac{{\sqrt[7]{7 (3 x^{2} + K)}}^{6}}{{({(7 (3 x^{2} + K))}^{\frac{1}{7}})}^{7}}$

Step 3.3.5.7.5.2

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

$y = - \frac{{\sqrt[7]{7 (3 x^{2} + K)}}^{6}}{{(7 (3 x^{2} + K))}^{\frac{1}{7} \cdot 7}}$

Step 3.3.5.7.5.3

Combine $\frac{1}{7}$ and $7$ .

$y = - \frac{{\sqrt[7]{7 (3 x^{2} + K)}}^{6}}{{(7 (3 x^{2} + K))}^{\frac{7}{7}}}$

Step 3.3.5.7.5.4

Cancel the common factor of $7$ .

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Step 3.3.5.7.5.4.1

Cancel the common factor.

$y = - \frac{{\sqrt[7]{7 (3 x^{2} + K)}}^{6}}{{(7 (3 x^{2} + K))}^{\frac{7}{7}}}$

Step 3.3.5.7.5.4.2

Rewrite the expression.

$y = - \frac{{\sqrt[7]{7 (3 x^{2} + K)}}^{6}}{{(7 (3 x^{2} + K))}^{1}}$

Step 3.3.5.7.5.5

Simplify.

$y = - \frac{{\sqrt[7]{7 (3 x^{2} + K)}}^{6}}{7 (3 x^{2} + K)}$

Step 3.3.5.8

Simplify the numerator.

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Step 3.3.5.8.1

Rewrite ${\sqrt[7]{7 (3 x^{2} + K)}}^{6}$ as $\sqrt[7]{{(7 (3 x^{2} + K))}^{6}}$ .

$y = - \frac{\sqrt[7]{{(7 (3 x^{2} + K))}^{6}}}{7 (3 x^{2} + K)}$

Step 3.3.5.8.2

Apply the product rule to $7 (3 x^{2} + K)$ .

$y = - \frac{\sqrt[7]{7^{6} {(3 x^{2} + K)}^{6}}}{7 (3 x^{2} + K)}$

Step 3.3.5.8.3

Raise $7$ to the power of $6$ .

$y = - \frac{\sqrt[7]{117649 {(3 x^{2} + K)}^{6}}}{7 (3 x^{2} + K)}$