Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

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

Step 1.2.2

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

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

Step 1.2.3

Combine and simplify the denominator.

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Step 1.2.3.1

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

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

Step 1.2.3.2

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

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

Step 1.2.3.3

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

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

Step 1.2.3.4

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

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

Step 1.2.3.5

Add $1$ and $1$ .

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

Step 1.2.3.6

Rewrite ${\sqrt{y - 1}}^{2}$ as $y - 1$ .

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Step 1.2.3.6.1

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

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

Step 1.2.3.6.2

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

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

Step 1.2.3.6.3

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

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

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.

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

Step 1.2.3.6.4.2

Rewrite the expression.

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

Step 1.2.3.6.5

Simplify.

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

Step 1.2.4

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

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

Step 1.2.5

Multiply $\frac{2 \sqrt{y - 1}}{y - 1} (x \sqrt{y - 1})$ .

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Step 1.2.5.1

Combine $x$ and $\frac{2 \sqrt{y - 1}}{y - 1}$ .

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

Step 1.2.5.2

Combine $\frac{x (2 \sqrt{y - 1})}{y - 1}$ and $\sqrt{y - 1}$ .

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

Step 1.2.5.3

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

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

Step 1.2.5.4

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

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

Step 1.2.5.5

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

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

Step 1.2.5.6

Add $1$ and $1$ .

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

Step 1.2.6

Rewrite ${\sqrt{y - 1}}^{2}$ as $y - 1$ .

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Step 1.2.6.1

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

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

Step 1.2.6.2

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

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

Step 1.2.6.3

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

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

Step 1.2.6.4

Cancel the common factor of $2$ .

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Step 1.2.6.4.1

Cancel the common factor.

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

Step 1.2.6.4.2

Rewrite the expression.

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

Step 1.2.6.5

Simplify.

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

Step 1.2.7

Cancel the common factor of $y - 1$ .

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Step 1.2.7.1

Cancel the common factor.

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

Step 1.2.7.2

Divide $x \cdot 2$ by $1$ .

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

Step 1.2.8

Move $2$ to the left of $x$ .

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u = y - 1$ . Then $d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.1.1

Let $u = y - 1$ . Find $\frac{d u}{d y}$ .

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Step 2.2.1.1.1

Differentiate $y - 1$ .

$\frac{d}{d y} [y - 1]$

Step 2.2.1.1.2

By the Sum Rule, the derivative of $y - 1$ with respect to $y$ is $\frac{d}{d y} [y] + \frac{d}{d y} [- 1]$ .

$\frac{d}{d y} [y] + \frac{d}{d y} [- 1]$

Step 2.2.1.1.3

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

$1 + \frac{d}{d y} [- 1]$

Step 2.2.1.1.4

Since $- 1$ is constant with respect to $y$ , the derivative of $- 1$ with respect to $y$ is $0$ .

$1 + 0$

Step 2.2.1.1.5

Add $1$ and $0$ .

$1$

Step 2.2.1.2

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

$\int \frac{1}{\sqrt{u}} d u = \int 2 x d x$

Step 2.2.2

Apply basic rules of exponents.

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

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Step 2.2.2.3.1

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

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

Step 2.2.2.3.2

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

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

Step 2.2.2.3.3

Move the negative in front of the fraction.

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

Step 2.2.3

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

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

Step 2.2.4

Replace all occurrences of $u$ with $y - 1$ .

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

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.

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

Step 2.3.2

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

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

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

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

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

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

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

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

Step 2.3.3.2.2

Cancel the common factor of $2$ .

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Step 2.3.3.2.2.1

Cancel the common factor.

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

Step 2.3.3.2.2.2

Rewrite the expression.

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

Step 2.3.3.2.3

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor.

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

Step 3.1.2.2

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

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

Step 3.2

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

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

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

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Step 3.3.1.1.1

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

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Step 3.3.1.1.1.1

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

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

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.

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

Step 3.3.1.1.1.2.2

Rewrite the expression.

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

Step 3.3.1.1.2

Simplify.

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

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

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

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Step 3.3.2.1.1

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

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

Step 3.3.2.1.2

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

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Step 3.3.2.1.2.1

Apply the distributive property.

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

Step 3.3.2.1.2.2

Apply the distributive property.

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

Step 3.3.2.1.2.3

Apply the distributive property.

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

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.

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

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

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

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

Step 3.3.2.1.3.1.2.2

Add $2$ and $2$ .

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

Step 3.3.2.1.3.1.3

Multiply $2$ by $2$ .

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

Step 3.3.2.1.3.1.4

Combine.

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

Step 3.3.2.1.3.1.5

Multiply $2$ by $2$ .

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

Step 3.3.2.1.3.1.6

Combine.

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

Step 3.3.2.1.3.1.7

Multiply $2$ by $2$ .

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

Step 3.3.2.1.3.1.8

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

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Step 3.3.2.1.3.1.8.1

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

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

Step 3.3.2.1.3.1.8.2

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

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

Step 3.3.2.1.3.1.8.3

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

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

Step 3.3.2.1.3.1.8.4

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

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

Step 3.3.2.1.3.1.8.5

Add $1$ and $1$ .

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

Step 3.3.2.1.3.1.8.6

Multiply $2$ by $2$ .

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

Step 3.3.2.1.3.2

Add $\frac{x^{2} K}{4}$ and $\frac{K x^{2}}{4}$ .

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Step 3.3.2.1.3.2.1

Reorder $x^{2}$ and $K$ .

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

Step 3.3.2.1.3.2.2

Add $\frac{K x^{2}}{4}$ and $\frac{K x^{2}}{4}$ .

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

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 $4$ .

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

Step 3.3.2.1.4.2

Cancel the common factor.

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

Step 3.3.2.1.4.3

Rewrite the expression.

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

Step 3.4

Add $1$ to both sides of the equation.

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

Step 4

Simplify the constant of integration.

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