Calculus Examples

$\frac{d y}{d x} = \frac{6 x - x^{3}}{2 y}$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Multiply both sides by $y$ .

$y \frac{d y}{d x} = y \frac{6 x - x^{3}}{2 y}$

Step 1.2

Simplify.

Tap for more steps...

Step 1.2.1

Cancel the common factor of $y$ .

Tap for more steps...

Step 1.2.1.1

Factor $y$ out of $2 y$ .

$y \frac{d y}{d x} = y \frac{6 x - x^{3}}{y \cdot 2}$

Step 1.2.1.2

Cancel the common factor.

$y \frac{d y}{d x} = y \frac{6 x - x^{3}}{y \cdot 2}$

Step 1.2.1.3

Rewrite the expression.

$y \frac{d y}{d x} = \frac{6 x - x^{3}}{2}$

Step 1.2.2

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

Tap for more steps...

Step 1.2.2.1

Factor $x$ out of $6 x$ .

$y \frac{d y}{d x} = \frac{x \cdot 6 - x^{3}}{2}$

Step 1.2.2.2

Factor $x$ out of $- x^{3}$ .

$y \frac{d y}{d x} = \frac{x \cdot 6 + x (- x^{2})}{2}$

Step 1.2.2.3

Factor $x$ out of $x \cdot 6 + x (- x^{2})$ .

$y \frac{d y}{d x} = \frac{x (6 - x^{2})}{2}$

Step 1.3

Rewrite the equation.

$y d y = \frac{x (6 - x^{2})}{2} d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int y d y = \int \frac{x (6 - x^{2})}{2} d x$

Step 2.2

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

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

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

Let $u = 6 - x^{2}$ . Then $d u = - 2 x d x$ , so $- \frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 2.3.2.1

Let $u = 6 - x^{2}$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 2.3.2.1.1

Differentiate $6 - x^{2}$ .

$\frac{d}{d x} [6 - x^{2}]$

Step 2.3.2.1.2

Differentiate.

Tap for more steps...

Step 2.3.2.1.2.1

By the Sum Rule, the derivative of $6 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [6] + \frac{d}{d x} [- x^{2}]$ .

$\frac{d}{d x} [6] + \frac{d}{d x} [- x^{2}]$

Step 2.3.2.1.2.2

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- x^{2}]$

Step 2.3.2.1.3

Evaluate $\frac{d}{d x} [- x^{2}]$ .

Tap for more steps...

Step 2.3.2.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$0 - \frac{d}{d x} [x^{2}]$

Step 2.3.2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$0 - (2 x)$

Step 2.3.2.1.3.3

Multiply $2$ by $- 1$ .

$0 - 2 x$

Step 2.3.2.1.4

Subtract $2 x$ from $0$ .

$- 2 x$

Step 2.3.2.2

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

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

Step 2.3.3

Simplify.

Tap for more steps...

Step 2.3.3.1

Move the negative in front of the fraction.

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

Step 2.3.3.2

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

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

Step 2.3.4

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

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

Step 2.3.5

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

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

Step 2.3.6

Simplify.

Tap for more steps...

Step 2.3.6.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

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

Step 2.3.6.2

Multiply $2$ by $2$ .

$\frac{1}{2} y^{2} + C_{1} = - \frac{1}{4} \int u d u$

Step 2.3.7

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

$\frac{1}{2} y^{2} + C_{1} = - \frac{1}{4} (\frac{1}{2} u^{2} + C_{2})$

Step 2.3.8

Simplify.

Tap for more steps...

Step 2.3.8.1

Rewrite $- \frac{1}{4} (\frac{1}{2} u^{2} + C_{2})$ as $- \frac{1}{4} \cdot \frac{1}{2} u^{2} + C_{2}$ .

$\frac{1}{2} y^{2} + C_{1} = - \frac{1}{4} \cdot \frac{1}{2} u^{2} + C_{2}$

Step 2.3.8.2

Simplify.

Tap for more steps...

Step 2.3.8.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{4}$ .

$\frac{1}{2} y^{2} + C_{1} = - \frac{1}{2 \cdot 4} u^{2} + C_{2}$

Step 2.3.8.2.2

Multiply $2$ by $4$ .

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

Step 2.3.9

Replace all occurrences of $u$ with $6 - x^{2}$ .

$\frac{1}{2} y^{2} + C_{1} = - \frac{1}{8} {(6 - x^{2})}^{2} + C_{2}$

Step 2.4

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

$\frac{1}{2} y^{2} = - \frac{1}{8} {(6 - x^{2})}^{2} + K$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} y^{2}) = 2 (- \frac{1}{8} {(6 - x^{2})}^{2} + K)$

Step 3.2

Simplify both sides of the equation.

Tap for more steps...

Step 3.2.1

Simplify the left side.

Tap for more steps...

Step 3.2.1.1

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

Tap for more steps...

Step 3.2.1.1.1

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

$2 \frac{y^{2}}{2} = 2 (- \frac{1}{8} {(6 - x^{2})}^{2} + K)$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.1.2.1

Cancel the common factor.

$2 \frac{y^{2}}{2} = 2 (- \frac{1}{8} {(6 - x^{2})}^{2} + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (- \frac{1}{8} {(6 - x^{2})}^{2} + K)$

Step 3.2.2

Simplify the right side.

Tap for more steps...

Step 3.2.2.1

Simplify $2 (- \frac{1}{8} {(6 - x^{2})}^{2} + K)$ .

Tap for more steps...

Step 3.2.2.1.1

Combine ${(6 - x^{2})}^{2}$ and $\frac{1}{8}$ .

$y^{2} = 2 (- \frac{{(6 - x^{2})}^{2}}{8} + K)$

Step 3.2.2.1.2

To write $K$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$y^{2} = 2 (- \frac{{(6 - x^{2})}^{2}}{8} + K \cdot \frac{8}{8})$

Step 3.2.2.1.3

Simplify terms.

Tap for more steps...

Step 3.2.2.1.3.1

Combine $K$ and $\frac{8}{8}$ .

$y^{2} = 2 (- \frac{{(6 - x^{2})}^{2}}{8} + \frac{K \cdot 8}{8})$

Step 3.2.2.1.3.2

Combine the numerators over the common denominator.

$y^{2} = 2 \frac{- {(6 - x^{2})}^{2} + K \cdot 8}{8}$

Step 3.2.2.1.3.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.2.1.3.3.1

Factor $2$ out of $8$ .

$y^{2} = 2 \frac{- {(6 - x^{2})}^{2} + K \cdot 8}{2 (4)}$

Step 3.2.2.1.3.3.2

Cancel the common factor.

$y^{2} = 2 \frac{- {(6 - x^{2})}^{2} + K \cdot 8}{2 \cdot 4}$

Step 3.2.2.1.3.3.3

Rewrite the expression.

$y^{2} = \frac{- {(6 - x^{2})}^{2} + K \cdot 8}{4}$

Step 3.2.2.1.4

Move $8$ to the left of $K$ .

$y^{2} = \frac{- {(6 - x^{2})}^{2} + 8 K}{4}$

Step 3.2.2.1.5

Simplify with factoring out.

Tap for more steps...

Step 3.2.2.1.5.1

Factor $- 1$ out of $- {(6 - x^{2})}^{2}$ .

$y^{2} = \frac{- ({(6 - x^{2})}^{2}) + 8 K}{4}$

Step 3.2.2.1.5.2

Factor $- 1$ out of $8 K$ .

$y^{2} = \frac{- ({(6 - x^{2})}^{2}) - (- 8 K)}{4}$

Step 3.2.2.1.5.3

Factor $- 1$ out of $- ({(6 - x^{2})}^{2}) - (- 8 K)$ .

$y^{2} = \frac{- ({(6 - x^{2})}^{2} - 8 K)}{4}$

Step 3.2.2.1.5.4

Simplify the expression.

Tap for more steps...

Step 3.2.2.1.5.4.1

Rewrite $- ({(6 - x^{2})}^{2} - 8 K)$ as $- 1 ({(6 - x^{2})}^{2} - 8 K)$ .

$y^{2} = \frac{- 1 ({(6 - x^{2})}^{2} - 8 K)}{4}$

Step 3.2.2.1.5.4.2

Move the negative in front of the fraction.

$y^{2} = - \frac{{(6 - x^{2})}^{2} - 8 K}{4}$

Step 3.3

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

$y = \pm \sqrt{- \frac{{(6 - x^{2})}^{2} - 8 K}{4}}$

Step 3.4

Simplify $\pm \sqrt{- \frac{{(6 - x^{2})}^{2} - 8 K}{4}}$ .

Tap for more steps...

Step 3.4.1

Rewrite ${(6 - x^{2})}^{2}$ as $(6 - x^{2}) (6 - x^{2})$ .

$y = \pm \sqrt{- \frac{(6 - x^{2}) (6 - x^{2}) - 8 K}{4}}$

Step 3.4.2

Expand $(6 - x^{2}) (6 - x^{2})$ using the FOIL Method.

Tap for more steps...

Step 3.4.2.1

Apply the distributive property.

$y = \pm \sqrt{- \frac{6 (6 - x^{2}) - x^{2} (6 - x^{2}) - 8 K}{4}}$

Step 3.4.2.2

Apply the distributive property.

$y = \pm \sqrt{- \frac{6 \cdot 6 + 6 (- x^{2}) - x^{2} (6 - x^{2}) - 8 K}{4}}$

Step 3.4.2.3

Apply the distributive property.

$y = \pm \sqrt{- \frac{6 \cdot 6 + 6 (- x^{2}) - x^{2} \cdot 6 - x^{2} (- x^{2}) - 8 K}{4}}$

Step 3.4.3

Simplify and combine like terms.

Tap for more steps...

Step 3.4.3.1

Simplify each term.

Tap for more steps...

Step 3.4.3.1.1

Multiply $6$ by $6$ .

$y = \pm \sqrt{- \frac{36 + 6 (- x^{2}) - x^{2} \cdot 6 - x^{2} (- x^{2}) - 8 K}{4}}$

Step 3.4.3.1.2

Multiply $- 1$ by $6$ .

$y = \pm \sqrt{- \frac{36 - 6 x^{2} - x^{2} \cdot 6 - x^{2} (- x^{2}) - 8 K}{4}}$

Step 3.4.3.1.3

Multiply $6$ by $- 1$ .

$y = \pm \sqrt{- \frac{36 - 6 x^{2} - 6 x^{2} - x^{2} (- x^{2}) - 8 K}{4}}$

Step 3.4.3.1.4

Rewrite using the commutative property of multiplication.

$y = \pm \sqrt{- \frac{36 - 6 x^{2} - 6 x^{2} - 1 \cdot - 1 x^{2} x^{2} - 8 K}{4}}$

Step 3.4.3.1.5

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 3.4.3.1.5.1

Move $x^{2}$ .

$y = \pm \sqrt{- \frac{36 - 6 x^{2} - 6 x^{2} - 1 \cdot - 1 (x^{2} x^{2}) - 8 K}{4}}$

Step 3.4.3.1.5.2

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

$y = \pm \sqrt{- \frac{36 - 6 x^{2} - 6 x^{2} - 1 \cdot - 1 x^{2 + 2} - 8 K}{4}}$

Step 3.4.3.1.5.3

Add $2$ and $2$ .

$y = \pm \sqrt{- \frac{36 - 6 x^{2} - 6 x^{2} - 1 \cdot - 1 x^{4} - 8 K}{4}}$

Step 3.4.3.1.6

Multiply $- 1$ by $- 1$ .

$y = \pm \sqrt{- \frac{36 - 6 x^{2} - 6 x^{2} + 1 x^{4} - 8 K}{4}}$

Step 3.4.3.1.7

Multiply $x^{4}$ by $1$ .

$y = \pm \sqrt{- \frac{36 - 6 x^{2} - 6 x^{2} + x^{4} - 8 K}{4}}$

Step 3.4.3.2

Subtract $6 x^{2}$ from $- 6 x^{2}$ .

$y = \pm \sqrt{- \frac{36 - 12 x^{2} + x^{4} - 8 K}{4}}$

Step 3.4.4

Rewrite $- \frac{36 - 12 x^{2} + x^{4} - 8 K}{4}$ as ${(\frac{1^{2}}{2})}^{2} \cdot (- (36 - 12 x^{2} + x^{4} - 8 K))$ .

Tap for more steps...

Step 3.4.4.1

Factor the perfect power $1^{2}$ out of $36 - 12 x^{2} + x^{4} - 8 K$ .

$y = \pm \sqrt{- \frac{1^{2} (36 - 12 x^{2} + x^{4} - 8 K)}{4}}$

Step 3.4.4.2

Factor the perfect power $2^{2}$ out of $4$ .

$y = \pm \sqrt{- \frac{1^{2} (36 - 12 x^{2} + x^{4} - 8 K)}{2^{2} \cdot 1}}$

Step 3.4.4.3

Rearrange the fraction $\frac{1^{2} (36 - 12 x^{2} + x^{4} - 8 K)}{2^{2} \cdot 1}$ .

$y = \pm \sqrt{- ({(\frac{1}{2})}^{2} (36 - 12 x^{2} + x^{4} - 8 K))}$

Step 3.4.4.4

Reorder $- 1$ and ${(\frac{1}{2})}^{2}$ .

$y = \pm \sqrt{{(\frac{1}{2})}^{2} \cdot - 1 (36 - 12 x^{2} + x^{4} - 8 K)}$

Step 3.4.4.5

Rewrite $1$ as $1^{2}$ .

$y = \pm \sqrt{{(\frac{1^{2}}{2})}^{2} \cdot - 1 (36 - 12 x^{2} + x^{4} - 8 K)}$

Step 3.4.4.6

Add parentheses.

$y = \pm \sqrt{{(\frac{1^{2}}{2})}^{2} \cdot (- (36 - 12 x^{2} + x^{4} - 8 K))}$

Step 3.4.5

Pull terms out from under the radical.

$y = \pm \frac{1^{2}}{2} \sqrt{- (36 - 12 x^{2} + x^{4} - 8 K)}$

Step 3.4.6

One to any power is one.

$y = \pm \frac{1}{2} \sqrt{- (36 - 12 x^{2} + x^{4} - 8 K)}$

Step 3.4.7

Combine $\frac{1}{2}$ and $\sqrt{- (36 - 12 x^{2} + x^{4} - 8 K)}$ .

$y = \pm \frac{\sqrt{- (36 - 12 x^{2} + x^{4} - 8 K)}}{2}$

Step 3.5

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

Tap for more steps...

Step 3.5.1

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

$y = \frac{\sqrt{- (36 - 12 x^{2} + x^{4} - 8 K)}}{2}$

Step 3.5.2

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

$y = - \frac{\sqrt{- (36 - 12 x^{2} + x^{4} - 8 K)}}{2}$

Step 3.5.3

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

$y = \frac{\sqrt{- (36 - 12 x^{2} + x^{4} - 8 K)}}{2}$

$y = - \frac{\sqrt{- (36 - 12 x^{2} + x^{4} - 8 K)}}{2}$

$y = \frac{\sqrt{- (36 - 12 x^{2} + x^{4} - 8 K)}}{2}$

$y = - \frac{\sqrt{- (36 - 12 x^{2} + x^{4} - 8 K)}}{2}$

$y = \frac{\sqrt{- (36 - 12 x^{2} + x^{4} - 8 K)}}{2}$

$y = - \frac{\sqrt{- (36 - 12 x^{2} + x^{4} - 8 K)}}{2}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt{- (36 - 12 x^{2} + x^{4} + K)}}{2}$

$y = - \frac{\sqrt{- (36 - 12 x^{2} + x^{4} + K)}}{2}$