Calculus Examples

$\frac{d y}{d x} = \frac{y}{\sqrt{x}}$

Step 1

Separate the variables.

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Step 1.1

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

$\frac{1}{y} \frac{d y}{d x} = \frac{1}{y} \cdot \frac{y}{\sqrt{x}}$

Step 1.2

Simplify.

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Step 1.2.1

Cancel the common factor of $y$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{1}{y} \frac{d y}{d x} = \frac{1}{y} \cdot \frac{y}{\sqrt{x}}$

Step 1.2.1.2

Rewrite the expression.

$\frac{1}{y} \frac{d y}{d x} = \frac{1}{\sqrt{x}}$

Step 1.2.2

Multiply $\frac{1}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{1}{y} \frac{d y}{d x} = \frac{1}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Step 1.2.3

Combine and simplify the denominator.

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Step 1.2.3.1

Multiply $\frac{1}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{1}{y} \frac{d y}{d x} = \frac{\sqrt{x}}{\sqrt{x} \sqrt{x}}$

Step 1.2.3.2

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

$\frac{1}{y} \frac{d y}{d x} = \frac{\sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}}$

Step 1.2.3.3

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

$\frac{1}{y} \frac{d y}{d x} = \frac{\sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}}$

Step 1.2.3.4

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

$\frac{1}{y} \frac{d y}{d x} = \frac{\sqrt{x}}{{\sqrt{x}}^{1 + 1}}$

Step 1.2.3.5

Add $1$ and $1$ .

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

Step 1.2.3.6

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

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Step 1.2.3.6.1

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

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

Step 1.2.3.6.2

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

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

Step 1.2.3.6.3

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

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

Step 1.2.3.6.4

Cancel the common factor of $2$ .

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Step 1.2.3.6.4.1

Cancel the common factor.

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

Step 1.2.3.6.4.2

Rewrite the expression.

$\frac{1}{y} \frac{d y}{d x} = \frac{\sqrt{x}}{x^{1}}$

Step 1.2.3.6.5

Simplify.

$\frac{1}{y} \frac{d y}{d x} = \frac{\sqrt{x}}{x}$

Step 1.3

Rewrite the equation.

$\frac{1}{y} d y = \frac{\sqrt{x}}{x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y} d y = \int \frac{\sqrt{x}}{x} d x$

Step 2.2

The integral of $\frac{1}{y}$ with respect to $y$ is $\ln (| y |)$ .

$\ln (| y |) + C_{1} = \int \frac{\sqrt{x}}{x} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Simplify the expression.

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Step 2.3.1.1

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

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

Step 2.3.1.2

Split the fraction into multiple fractions.

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Step 2.3.1.2.1

Move $x^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 2.3.1.2.2

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 2.3.1.2.2.1

Multiply $x$ by $x^{- \frac{1}{2}}$ .

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Step 2.3.1.2.2.1.1

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

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

Step 2.3.1.2.2.1.2

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

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

Step 2.3.1.2.2.2

Write $1$ as a fraction with a common denominator.

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

Step 2.3.1.2.2.3

Combine the numerators over the common denominator.

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

Step 2.3.1.2.2.4

Subtract $1$ from $2$ .

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

Step 2.3.1.3

Apply basic rules of exponents.

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Step 2.3.1.3.1

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

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

Step 2.3.1.3.2

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

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Step 2.3.1.3.2.1

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

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

Step 2.3.1.3.2.2

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

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

Step 2.3.1.3.2.3

Move the negative in front of the fraction.

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

Step 2.3.2

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

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

Step 2.4

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

$\ln (| y |) = 2 x^{\frac{1}{2}} + C$

Step 3

Solve for $y$ .

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Step 3.1

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (| y |)} = e^{2 x^{\frac{1}{2}} + C}$

Step 3.2

Rewrite $\ln (| y |) = 2 x^{\frac{1}{2}} + C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{2 x^{\frac{1}{2}} + C} = | y |$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Rewrite the equation as $| y | = e^{2 x^{\frac{1}{2}} + C}$ .

$| y | = e^{2 x^{\frac{1}{2}} + C}$

Step 3.3.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$y = \pm e^{2 x^{\frac{1}{2}} + C}$

Step 4

Group the constant terms together.

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Step 4.1

Rewrite $e^{2 x^{\frac{1}{2}} + C}$ as $e^{2 x^{\frac{1}{2}}} e^{C}$ .

$y = \pm e^{2 x^{\frac{1}{2}}} e^{C}$

Step 4.2

Reorder $e^{2 x^{\frac{1}{2}}}$ and $e^{C}$ .

$y = \pm e^{C} e^{2 x^{\frac{1}{2}}}$

Step 4.3

Combine constants with the plus or minus.

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