Calculus Examples

$x^{2} \frac{d w}{d x} = \sqrt{w} (4 x + 2)$

Step 1

Separate the variables.

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Step 1.1

Divide each term in $x^{2} \frac{d w}{d x} = \sqrt{w} (4 x + 2)$ by $x^{2}$ and simplify.

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Step 1.1.1

Divide each term in $x^{2} \frac{d w}{d x} = \sqrt{w} (4 x + 2)$ by $x^{2}$ .

$\frac{x^{2} \frac{d w}{d x}}{x^{2}} = \frac{\sqrt{w} (4 x + 2)}{x^{2}}$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

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

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{x^{2} \frac{d w}{d x}}{x^{2}} = \frac{\sqrt{w} (4 x + 2)}{x^{2}}$

Step 1.1.2.1.2

Divide $\frac{d w}{d x}$ by $1$ .

$\frac{d w}{d x} = \frac{\sqrt{w} (4 x + 2)}{x^{2}}$

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Factor $2$ out of $4 x + 2$ .

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Step 1.1.3.1.1

Factor $2$ out of $4 x$ .

$\frac{d w}{d x} = \frac{\sqrt{w} (2 (2 x) + 2)}{x^{2}}$

Step 1.1.3.1.2

Factor $2$ out of $2$ .

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

Step 1.1.3.1.3

Factor $2$ out of $2 (2 x) + 2 (1)$ .

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

$\frac{d w}{d x} = \frac{\sqrt{w} \cdot 2 (2 x + 1)}{x^{2}}$

Step 1.1.3.2

Move $2$ to the left of $\sqrt{w}$ .

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

Step 1.2

Regroup factors.

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

Step 1.3

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

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

Step 1.4

Simplify.

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Step 1.4.1

Rewrite using the commutative property of multiplication.

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

Step 1.4.2

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

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = 2 (\frac{1}{\sqrt{w}} \cdot \frac{\sqrt{w}}{\sqrt{w}}) (\sqrt{w} \frac{2 x + 1}{x^{2}})$

Step 1.4.3

Combine and simplify the denominator.

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Step 1.4.3.1

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

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

Step 1.4.3.2

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

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

Step 1.4.3.3

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

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

Step 1.4.3.4

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

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

Step 1.4.3.5

Add $1$ and $1$ .

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

Step 1.4.3.6

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

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Step 1.4.3.6.1

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

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

Step 1.4.3.6.2

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

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = 2 \frac{\sqrt{w}}{w^{\frac{1}{2} \cdot 2}} (\sqrt{w} \frac{2 x + 1}{x^{2}})$

Step 1.4.3.6.3

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

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

Step 1.4.3.6.4

Cancel the common factor of $2$ .

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Step 1.4.3.6.4.1

Cancel the common factor.

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

Step 1.4.3.6.4.2

Rewrite the expression.

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

Step 1.4.3.6.5

Simplify.

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

Step 1.4.4

Combine $2$ and $\frac{\sqrt{w}}{w}$ .

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

Step 1.4.5

Combine $\sqrt{w}$ and $\frac{2 x + 1}{x^{2}}$ .

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = \frac{2 \sqrt{w}}{w} \cdot \frac{\sqrt{w} (2 x + 1)}{x^{2}}$

Step 1.4.6

Combine.

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

Step 1.4.7

Simplify the numerator.

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Step 1.4.7.1

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

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

Step 1.4.7.2

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

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

Step 1.4.7.3

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

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

Step 1.4.7.4

Add $1$ and $1$ .

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

Step 1.4.8

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

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Step 1.4.8.1

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

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

Step 1.4.8.2

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

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = \frac{2 w^{\frac{1}{2} \cdot 2} (2 x + 1)}{w x^{2}}$

Step 1.4.8.3

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

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

Step 1.4.8.4

Cancel the common factor of $2$ .

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Step 1.4.8.4.1

Cancel the common factor.

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

Step 1.4.8.4.2

Rewrite the expression.

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

Step 1.4.8.5

Simplify.

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

Step 1.4.9

Cancel the common factor of $w$ .

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Step 1.4.9.1

Cancel the common factor.

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

Step 1.4.9.2

Rewrite the expression.

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

Step 1.5

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{\sqrt{w}} d w = \int \frac{2 (2 x + 1)}{x^{2}} d x$

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 $\sqrt{w}$ as $w^{\frac{1}{2}}$ .

$\int \frac{1}{w^{\frac{1}{2}}} d w = \int \frac{2 (2 x + 1)}{x^{2}} d x$

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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Step 2.2.1.3.1

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

$\int w^{\frac{1}{2} \cdot - 1} d w = \int \frac{2 (2 x + 1)}{x^{2}} d x$

Step 2.2.1.3.2

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

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

Step 2.2.1.3.3

Move the negative in front of the fraction.

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

Step 2.2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 2.3.2

Apply basic rules of exponents.

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Step 2.3.2.1

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

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

Step 2.3.2.2

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

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Step 2.3.2.2.1

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

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

Step 2.3.2.2.2

Multiply $2$ by $- 1$ .

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

Step 2.3.3

Multiply $(2 x + 1) x^{- 2}$ .

$2 w^{\frac{1}{2}} + C_{1} = 2 \int 2 x \cdot x^{- 2} + 1 x^{- 2} d x$

Step 2.3.4

Simplify.

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Step 2.3.4.1

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

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Step 2.3.4.1.1

Move $x^{- 2}$ .

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

Step 2.3.4.1.2

Multiply $x^{- 2}$ by $x$ .

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Step 2.3.4.1.2.1

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

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

Step 2.3.4.1.2.2

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

$2 w^{\frac{1}{2}} + C_{1} = 2 \int 2 x^{- 2 + 1} + 1 x^{- 2} d x$

Step 2.3.4.1.3

Add $- 2$ and $1$ .

$2 w^{\frac{1}{2}} + C_{1} = 2 \int 2 x^{- 1} + 1 x^{- 2} d x$

Step 2.3.4.2

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

$2 w^{\frac{1}{2}} + C_{1} = 2 \int 2 x^{- 1} + x^{- 2} d x$

Step 2.3.5

Split the single integral into multiple integrals.

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

Step 2.3.6

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 2.3.7

The integral of $x^{- 1}$ with respect to $x$ is $\ln (| x |)$ .

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

Step 2.3.8

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

$2 w^{\frac{1}{2}} + C_{1} = 2 (2 (\ln (| x |) + C_{2}) - x^{- 1} + C_{3})$

Step 2.3.9

Simplify.

$2 w^{\frac{1}{2}} + C_{1} = 2 (2 \ln (| x |) - \frac{1}{x}) + C_{4}$

Step 2.4

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

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

Step 3

Solve for $w$ .

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Step 3.1

Divide each term in $2 w^{\frac{1}{2}} = 2 (2 \ln (| x |) - \frac{1}{x}) + C$ by $2$ and simplify.

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Step 3.1.1

Divide each term in $2 w^{\frac{1}{2}} = 2 (2 \ln (| x |) - \frac{1}{x}) + C$ by $2$ .

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor.

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

Step 3.1.2.2

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

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

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.

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

Step 3.1.3.1.2

Divide $2 \ln (| x |) - \frac{1}{x}$ by $1$ .

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

Step 3.2

Simplify $2 \ln (| x |)$ by moving $2$ inside the logarithm.

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

Step 3.3

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

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

Step 3.4

Simplify the exponent.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

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

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Step 3.4.1.1.1

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

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Step 3.4.1.1.1.1

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

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

Step 3.4.1.1.1.2

Cancel the common factor of $2$ .

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Step 3.4.1.1.1.2.1

Cancel the common factor.

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

Step 3.4.1.1.1.2.2

Rewrite the expression.

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

Step 3.4.1.1.2

Simplify.

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

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Remove the absolute value in ${| x |}^{2}$ because exponentiations with even powers are always positive.

$w = {(\ln (x^{2}) - \frac{1}{x} + \frac{C}{2})}^{2}$

Step 4

Simplify the constant of integration.

$w = {(\ln (x^{2}) - \frac{1}{x} + C)}^{2}$