Calculus Examples

$\frac{d y}{d x} = 9 x y^{\frac{1}{2}}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{y^{\frac{1}{2}}}$ .

$\frac{1}{y^{\frac{1}{2}}} \frac{d y}{d x} = \frac{1}{y^{\frac{1}{2}}} (9 x y^{\frac{1}{2}})$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

$\frac{1}{y^{\frac{1}{2}}} \frac{d y}{d x} = 9 \frac{1}{y^{\frac{1}{2}}} (x y^{\frac{1}{2}})$

Step 1.2.2

Combine $9$ and $\frac{1}{y^{\frac{1}{2}}}$ .

$\frac{1}{y^{\frac{1}{2}}} \frac{d y}{d x} = \frac{9}{y^{\frac{1}{2}}} (x y^{\frac{1}{2}})$

Step 1.2.3

Cancel the common factor of $y^{\frac{1}{2}}$ .

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Step 1.2.3.1

Factor $y^{\frac{1}{2}}$ out of $x y^{\frac{1}{2}}$ .

$\frac{1}{y^{\frac{1}{2}}} \frac{d y}{d x} = \frac{9}{y^{\frac{1}{2}}} (y^{\frac{1}{2}} x)$

Step 1.2.3.2

Cancel the common factor.

$\frac{1}{y^{\frac{1}{2}}} \frac{d y}{d x} = \frac{9}{y^{\frac{1}{2}}} (y^{\frac{1}{2}} x)$

Step 1.2.3.3

Rewrite the expression.

$\frac{1}{y^{\frac{1}{2}}} \frac{d y}{d x} = 9 x$

Step 1.3

Rewrite the equation.

$\frac{1}{y^{\frac{1}{2}}} d y = 9 x d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y^{\frac{1}{2}}} d y = \int 9 x d x$

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 $y^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$\int {(y^{\frac{1}{2}})}^{- 1} d y = \int 9 x d x$

Step 2.2.1.2

Multiply the exponents in ${(y^{\frac{1}{2}})}^{- 1}$ .

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Step 2.2.1.2.1

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

$\int y^{\frac{1}{2} \cdot - 1} d y = \int 9 x d x$

Step 2.2.1.2.2

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

$\int y^{\frac{- 1}{2}} d y = \int 9 x d x$

Step 2.2.1.2.3

Move the negative in front of the fraction.

$\int y^{- \frac{1}{2}} d y = \int 9 x d x$

Step 2.2.2

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

$2 y^{\frac{1}{2}} + C_{1} = \int 9 x d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$2 y^{\frac{1}{2}} + C_{1} = 9 \int x d x$

Step 2.3.2

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

$2 y^{\frac{1}{2}} + C_{1} = 9 (\frac{1}{2} x^{2} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $9 (\frac{1}{2} x^{2} + C_{2})$ as $9 (\frac{1}{2}) x^{2} + C_{2}$ .

$2 y^{\frac{1}{2}} + C_{1} = 9 (\frac{1}{2}) x^{2} + C_{2}$

Step 2.3.3.2

Combine $9$ and $\frac{1}{2}$ .

$2 y^{\frac{1}{2}} + C_{1} = \frac{9}{2} x^{2} + C_{2}$

Step 2.4

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

$2 y^{\frac{1}{2}} = \frac{9}{2} x^{2} + K$

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $2 y^{\frac{1}{2}} = \frac{9}{2} x^{2} + K$ by $2$ and simplify.

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Step 3.1.1

Divide each term in $2 y^{\frac{1}{2}} = \frac{9}{2} x^{2} + K$ by $2$ .

$\frac{2 y^{\frac{1}{2}}}{2} = \frac{\frac{9}{2} x^{2}}{2} + \frac{K}{2}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor.

$\frac{2 y^{\frac{1}{2}}}{2} = \frac{\frac{9}{2} x^{2}}{2} + \frac{K}{2}$

Step 3.1.2.2

Divide $y^{\frac{1}{2}}$ by $1$ .

$y^{\frac{1}{2}} = \frac{\frac{9}{2} x^{2}}{2} + \frac{K}{2}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Combine $\frac{9}{2}$ and $x^{2}$ .

$y^{\frac{1}{2}} = \frac{\frac{9 x^{2}}{2}}{2} + \frac{K}{2}$

Step 3.1.3.1.2

Multiply the numerator by the reciprocal of the denominator.

$y^{\frac{1}{2}} = \frac{9 x^{2}}{2} \cdot \frac{1}{2} + \frac{K}{2}$

Step 3.1.3.1.3

Combine.

$y^{\frac{1}{2}} = \frac{9 x^{2} \cdot 1}{2 \cdot 2} + \frac{K}{2}$

Step 3.1.3.1.4

Multiply $9$ by $1$ .

$y^{\frac{1}{2}} = \frac{9 x^{2}}{2 \cdot 2} + \frac{K}{2}$

Step 3.1.3.1.5

Multiply $2$ by $2$ .

$y^{\frac{1}{2}} = \frac{9 x^{2}}{4} + \frac{K}{2}$

Step 3.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(y^{\frac{1}{2}})}^{2} = {(\frac{9 x^{2}}{4} + \frac{K}{2})}^{2}$

Step 3.3

Simplify the exponent.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

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

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Step 3.3.1.1.1

Multiply the exponents in ${(y^{\frac{1}{2}})}^{2}$ .

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Step 3.3.1.1.1.1

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

$y^{\frac{1}{2} \cdot 2} = {(\frac{9 x^{2}}{4} + \frac{K}{2})}^{2}$

Step 3.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.1.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{2} \cdot 2} = {(\frac{9 x^{2}}{4} + \frac{K}{2})}^{2}$

Step 3.3.1.1.1.2.2

Rewrite the expression.

$y^{1} = {(\frac{9 x^{2}}{4} + \frac{K}{2})}^{2}$

Step 3.3.1.1.2

Simplify.

$y = {(\frac{9 x^{2}}{4} + \frac{K}{2})}^{2}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify ${(\frac{9 x^{2}}{4} + \frac{K}{2})}^{2}$ .

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Step 3.3.2.1.1

Rewrite ${(\frac{9 x^{2}}{4} + \frac{K}{2})}^{2}$ as $(\frac{9 x^{2}}{4} + \frac{K}{2}) (\frac{9 x^{2}}{4} + \frac{K}{2})$ .

$y = (\frac{9 x^{2}}{4} + \frac{K}{2}) (\frac{9 x^{2}}{4} + \frac{K}{2})$

Step 3.3.2.1.2

Expand $(\frac{9 x^{2}}{4} + \frac{K}{2}) (\frac{9 x^{2}}{4} + \frac{K}{2})$ using the FOIL Method.

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Step 3.3.2.1.2.1

Apply the distributive property.

$y = \frac{9 x^{2}}{4} (\frac{9 x^{2}}{4} + \frac{K}{2}) + \frac{K}{2} (\frac{9 x^{2}}{4} + \frac{K}{2})$

Step 3.3.2.1.2.2

Apply the distributive property.

$y = \frac{9 x^{2}}{4} \cdot \frac{9 x^{2}}{4} + \frac{9 x^{2}}{4} \cdot \frac{K}{2} + \frac{K}{2} (\frac{9 x^{2}}{4} + \frac{K}{2})$

Step 3.3.2.1.2.3

Apply the distributive property.

$y = \frac{9 x^{2}}{4} \cdot \frac{9 x^{2}}{4} + \frac{9 x^{2}}{4} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{9 x^{2}}{4} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3

Simplify and combine like terms.

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Step 3.3.2.1.3.1

Simplify each term.

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Step 3.3.2.1.3.1.1

Combine.

$y = \frac{9 x^{2} (9 x^{2})}{4 \cdot 4} + \frac{9 x^{2}}{4} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{9 x^{2}}{4} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.2

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

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Step 3.3.2.1.3.1.2.1

Move $x^{2}$ .

$y = \frac{9 (x^{2} x^{2}) \cdot 9}{4 \cdot 4} + \frac{9 x^{2}}{4} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{9 x^{2}}{4} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.2.2

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

$y = \frac{9 x^{2 + 2} \cdot 9}{4 \cdot 4} + \frac{9 x^{2}}{4} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{9 x^{2}}{4} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.2.3

Add $2$ and $2$ .

$y = \frac{9 x^{4} \cdot 9}{4 \cdot 4} + \frac{9 x^{2}}{4} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{9 x^{2}}{4} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.3

Multiply $9$ by $9$ .

$y = \frac{81 x^{4}}{4 \cdot 4} + \frac{9 x^{2}}{4} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{9 x^{2}}{4} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.4

Multiply $4$ by $4$ .

$y = \frac{81 x^{4}}{16} + \frac{9 x^{2}}{4} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{9 x^{2}}{4} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.5

Combine.

$y = \frac{81 x^{4}}{16} + \frac{9 x^{2} K}{4 \cdot 2} + \frac{K}{2} \cdot \frac{9 x^{2}}{4} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.6

Multiply $4$ by $2$ .

$y = \frac{81 x^{4}}{16} + \frac{9 x^{2} K}{8} + \frac{K}{2} \cdot \frac{9 x^{2}}{4} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.7

Combine.

$y = \frac{81 x^{4}}{16} + \frac{9 x^{2} K}{8} + \frac{K (9 x^{2})}{2 \cdot 4} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.8

Multiply $2$ by $4$ .

$y = \frac{81 x^{4}}{16} + \frac{9 x^{2} K}{8} + \frac{K \cdot 9 x^{2}}{8} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.9

Move $9$ to the left of $K$ .

$y = \frac{81 x^{4}}{16} + \frac{9 x^{2} K}{8} + \frac{9 \cdot K x^{2}}{8} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.10

Multiply $\frac{K}{2} \cdot \frac{K}{2}$ .

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Step 3.3.2.1.3.1.10.1

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

$y = \frac{81 x^{4}}{16} + \frac{9 x^{2} K}{8} + \frac{9 K x^{2}}{8} + \frac{K \cdot K}{2 \cdot 2}$

Step 3.3.2.1.3.1.10.2

Raise $K$ to the power of $1$ .

$y = \frac{81 x^{4}}{16} + \frac{9 x^{2} K}{8} + \frac{9 K x^{2}}{8} + \frac{K^{1} K}{2 \cdot 2}$

Step 3.3.2.1.3.1.10.3

Raise $K$ to the power of $1$ .

$y = \frac{81 x^{4}}{16} + \frac{9 x^{2} K}{8} + \frac{9 K x^{2}}{8} + \frac{K^{1} K^{1}}{2 \cdot 2}$

Step 3.3.2.1.3.1.10.4

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

$y = \frac{81 x^{4}}{16} + \frac{9 x^{2} K}{8} + \frac{9 K x^{2}}{8} + \frac{K^{1 + 1}}{2 \cdot 2}$

Step 3.3.2.1.3.1.10.5

Add $1$ and $1$ .

$y = \frac{81 x^{4}}{16} + \frac{9 x^{2} K}{8} + \frac{9 K x^{2}}{8} + \frac{K^{2}}{2 \cdot 2}$

Step 3.3.2.1.3.1.10.6

Multiply $2$ by $2$ .

$y = \frac{81 x^{4}}{16} + \frac{9 x^{2} K}{8} + \frac{9 K x^{2}}{8} + \frac{K^{2}}{4}$

Step 3.3.2.1.3.2

Add $\frac{9 x^{2} K}{8}$ and $\frac{9 K x^{2}}{8}$ .

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Step 3.3.2.1.3.2.1

Move $x^{2}$ .

$y = \frac{81 x^{4}}{16} + \frac{9 K x^{2}}{8} + \frac{9 K x^{2}}{8} + \frac{K^{2}}{4}$

Step 3.3.2.1.3.2.2

Add $\frac{9 K x^{2}}{8}$ and $\frac{9 K x^{2}}{8}$ .

$y = \frac{81 x^{4}}{16} + 2 \frac{9 K x^{2}}{8} + \frac{K^{2}}{4}$

Step 3.3.2.1.4

Cancel the common factor of $2$ .

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Step 3.3.2.1.4.1

Factor $2$ out of $8$ .

$y = \frac{81 x^{4}}{16} + 2 \frac{9 K x^{2}}{2 (4)} + \frac{K^{2}}{4}$

Step 3.3.2.1.4.2

Cancel the common factor.

$y = \frac{81 x^{4}}{16} + 2 \frac{9 K x^{2}}{2 \cdot 4} + \frac{K^{2}}{4}$

Step 3.3.2.1.4.3

Rewrite the expression.

$y = \frac{81 x^{4}}{16} + \frac{9 K x^{2}}{4} + \frac{K^{2}}{4}$

Step 4

Simplify the constant of integration.

$y = \frac{81 x^{4}}{16} + \frac{K x^{2}}{4} + K$