Calculus Examples

$\frac{d y}{d x} = \frac{4 \sqrt{y} \ln (x)}{x}$

Step 1

Separate the variables.

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Step 1.1

Regroup factors.

$\frac{d y}{d x} = 4 \sqrt{y} \frac{\ln (x)}{x}$

Step 1.2

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{1}{\sqrt{y}} (4 \sqrt{y} \frac{\ln (x)}{x})$

Step 1.3

Simplify.

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Step 1.3.1

Rewrite using the commutative property of multiplication.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 4 \frac{1}{\sqrt{y}} (\sqrt{y} \frac{\ln (x)}{x})$

Step 1.3.2

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 4 (\frac{1}{\sqrt{y}} \cdot \frac{\sqrt{y}}{\sqrt{y}}) (\sqrt{y} \frac{\ln (x)}{x})$

Step 1.3.3

Combine and simplify the denominator.

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Step 1.3.3.1

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 4 \frac{\sqrt{y}}{\sqrt{y} \sqrt{y}} (\sqrt{y} \frac{\ln (x)}{x})$

Step 1.3.3.2

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 4 \frac{\sqrt{y}}{{\sqrt{y}}^{1} \sqrt{y}} (\sqrt{y} \frac{\ln (x)}{x})$

Step 1.3.3.3

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 4 \frac{\sqrt{y}}{{\sqrt{y}}^{1} {\sqrt{y}}^{1}} (\sqrt{y} \frac{\ln (x)}{x})$

Step 1.3.3.4

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 4 \frac{\sqrt{y}}{{\sqrt{y}}^{1 + 1}} (\sqrt{y} \frac{\ln (x)}{x})$

Step 1.3.3.5

Add $1$ and $1$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 4 \frac{\sqrt{y}}{{\sqrt{y}}^{2}} (\sqrt{y} \frac{\ln (x)}{x})$

Step 1.3.3.6

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

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Step 1.3.3.6.1

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 4 \frac{\sqrt{y}}{{(y^{\frac{1}{2}})}^{2}} (\sqrt{y} \frac{\ln (x)}{x})$

Step 1.3.3.6.2

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 4 \frac{\sqrt{y}}{y^{\frac{1}{2} \cdot 2}} (\sqrt{y} \frac{\ln (x)}{x})$

Step 1.3.3.6.3

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 4 \frac{\sqrt{y}}{y^{\frac{2}{2}}} (\sqrt{y} \frac{\ln (x)}{x})$

Step 1.3.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.3.6.4.1

Cancel the common factor.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 4 \frac{\sqrt{y}}{y^{\frac{2}{2}}} (\sqrt{y} \frac{\ln (x)}{x})$

Step 1.3.3.6.4.2

Rewrite the expression.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 4 \frac{\sqrt{y}}{y^{1}} (\sqrt{y} \frac{\ln (x)}{x})$

Step 1.3.3.6.5

Simplify.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 4 \frac{\sqrt{y}}{y} (\sqrt{y} \frac{\ln (x)}{x})$

Step 1.3.4

Combine $4$ and $\frac{\sqrt{y}}{y}$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{4 \sqrt{y}}{y} (\sqrt{y} \frac{\ln (x)}{x})$

Step 1.3.5

Combine $\sqrt{y}$ and $\frac{\ln (x)}{x}$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{4 \sqrt{y}}{y} \cdot \frac{\sqrt{y} \ln (x)}{x}$

Step 1.3.6

Multiply $\frac{4 \sqrt{y}}{y} \cdot \frac{\sqrt{y} \ln (x)}{x}$ .

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Step 1.3.6.1

Multiply $\frac{4 \sqrt{y}}{y}$ by $\frac{\sqrt{y} \ln (x)}{x}$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{4 \sqrt{y} (\sqrt{y} \ln (x))}{y x}$

Step 1.3.6.2

Reorder $\ln (x)$ and $4$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{4 \cdot \ln (x) \sqrt{y} \sqrt{y}}{y x}$

Step 1.3.6.3

Simplify $4 \cdot \ln (x)$ by moving $4$ inside the logarithm.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{\ln (x^{4}) \sqrt{y} \sqrt{y}}{y x}$

Step 1.3.6.4

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{\ln (x^{4}) ({\sqrt{y}}^{1} \sqrt{y})}{y x}$

Step 1.3.6.5

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{\ln (x^{4}) ({\sqrt{y}}^{1} {\sqrt{y}}^{1})}{y x}$

Step 1.3.6.6

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{\ln (x^{4}) {\sqrt{y}}^{1 + 1}}{y x}$

Step 1.3.6.7

Add $1$ and $1$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{\ln (x^{4}) {\sqrt{y}}^{2}}{y x}$

Step 1.3.7

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

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Step 1.3.7.1

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{\ln (x^{4}) {(y^{\frac{1}{2}})}^{2}}{y x}$

Step 1.3.7.2

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{\ln (x^{4}) y^{\frac{1}{2} \cdot 2}}{y x}$

Step 1.3.7.3

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{\ln (x^{4}) y^{\frac{2}{2}}}{y x}$

Step 1.3.7.4

Cancel the common factor of $2$ .

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Step 1.3.7.4.1

Cancel the common factor.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{\ln (x^{4}) y^{\frac{2}{2}}}{y x}$

Step 1.3.7.4.2

Rewrite the expression.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{\ln (x^{4}) y^{1}}{y x}$

Step 1.3.7.5

Simplify.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{\ln (x^{4}) y}{y x}$

Step 1.3.8

Expand $\ln (x^{4})$ by moving $4$ outside the logarithm.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{4 \ln (x) y}{y x}$

Step 1.3.9

Cancel the common factor of $y$ .

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Step 1.3.9.1

Cancel the common factor.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{4 \ln (x) y}{y x}$

Step 1.3.9.2

Rewrite the expression.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{4 \ln (x)}{x}$

Step 1.3.10

Simplify $4 \ln (x)$ by moving $4$ inside the logarithm.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{\ln (x^{4})}{x}$

Step 1.4

Rewrite the equation.

$\frac{1}{\sqrt{y}} d y = \frac{\ln (x^{4})}{x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{\sqrt{y}} d y = \int \frac{\ln (x^{4})}{x} d x$

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

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

$\int \frac{1}{y^{\frac{1}{2}}} d y = \int \frac{\ln (x^{4})}{x} d x$

Step 2.2.1.2

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

$\int {(y^{\frac{1}{2}})}^{- 1} d y = \int \frac{\ln (x^{4})}{x} d x$

Step 2.2.1.3

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

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Step 2.2.1.3.1

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

$\int y^{\frac{1}{2} \cdot - 1} d y = \int \frac{\ln (x^{4})}{x} d x$

Step 2.2.1.3.2

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

$\int y^{\frac{- 1}{2}} d y = \int \frac{\ln (x^{4})}{x} d x$

Step 2.2.1.3.3

Move the negative in front of the fraction.

$\int y^{- \frac{1}{2}} d y = \int \frac{\ln (x^{4})}{x} d x$

Step 2.2.2

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

$2 y^{\frac{1}{2}} + C_{1} = \int \frac{\ln (x^{4})}{x} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Rewrite $\frac{\ln (x^{4})}{x}$ as $\frac{4 \ln (x)}{x}$ .

$2 y^{\frac{1}{2}} + C_{1} = \int \frac{4 \ln (x)}{x} d x$

Step 2.3.2

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$2 y^{\frac{1}{2}} + C_{1} = 4 \int \frac{\ln (x)}{x} d x$

Step 2.3.3

Let $u = \ln (x)$ . Then $d u = \frac{1}{x} d x$ , so $x d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.3.1

Let $u = \ln (x)$ . Find $\frac{d u}{d x}$ .

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Step 2.3.3.1.1

Differentiate $\ln (x)$ .

$\frac{d}{d x} [\ln (x)]$

Step 2.3.3.1.2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$\frac{1}{x}$

Step 2.3.3.2

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

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

Step 2.3.4

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

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

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

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

Step 2.3.5.2

Simplify.

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Step 2.3.5.2.1

Combine $4$ and $\frac{1}{2}$ .

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

Step 2.3.5.2.2

Cancel the common factor of $4$ and $2$ .

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Step 2.3.5.2.2.1

Factor $2$ out of $4$ .

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

Step 2.3.5.2.2.2

Cancel the common factors.

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Step 2.3.5.2.2.2.1

Factor $2$ out of $2$ .

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

Step 2.3.5.2.2.2.2

Cancel the common factor.

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

Step 2.3.5.2.2.2.3

Rewrite the expression.

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

Step 2.3.5.2.2.2.4

Divide $2$ by $1$ .

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

Step 2.3.6

Replace all occurrences of $u$ with $\ln (x)$ .

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor.

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

Step 3.1.2.2

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

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

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Cancel the common factor of $2$ .

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Step 3.1.3.1.1

Cancel the common factor.

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

Step 3.1.3.1.2

Divide $\ln^{2} (x)$ by $1$ .

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

Step 3.2

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

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

Step 3.3

Simplify the left side.

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Step 3.3.1

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

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Step 3.3.1.1

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

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Step 3.3.1.1.1

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

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

Step 3.3.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.1.1.2.1

Cancel the common factor.

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

Step 3.3.1.1.2.2

Rewrite the expression.

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

Step 3.3.1.2

Simplify.

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

Step 4

Simplify the constant of integration.

$y = {(\ln^{2} (x) + K)}^{2}$