Calculus Examples

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

Step 1

Integrate both sides with respect to $x$ .

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Step 1.1

The first derivative is equal to the integral of the second derivative with respect to $x$ .

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

Step 1.2

Let $u_{1} = 2 x - 1$ . Then $d u_{1} = 2 d x$ , so $\frac{1}{2} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 1.2.1

Let $u_{1} = 2 x - 1$ . Find $\frac{d u_{1}}{d x}$ .

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Step 1.2.1.1

Differentiate $2 x - 1$ .

$\frac{d}{d x} [2 x - 1]$

Step 1.2.1.2

By the Sum Rule, the derivative of $2 x - 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$

Step 1.2.1.3

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 1.2.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.2.1.3.2

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

$2 \cdot 1 + \frac{d}{d x} [- 1]$

Step 1.2.1.3.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [- 1]$

Step 1.2.1.4

Differentiate using the Constant Rule.

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Step 1.2.1.4.1

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

$2 + 0$

Step 1.2.1.4.2

Add $2$ and $0$ .

$2$

Step 1.2.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

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

Step 1.3

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

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

Step 1.4

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , move $\frac{1}{2}$ out of the integral.

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

Step 1.5

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

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

Step 1.6

By the Power Rule, the integral of ${u_{1}}^{\frac{1}{2}}$ with respect to $u_{1}$ is $\frac{2}{3} {u_{1}}^{\frac{3}{2}}$ .

$\frac{d y}{d x} = \frac{1}{2} (\frac{2}{3} {u_{1}}^{\frac{3}{2}} + C_{1})$

Step 1.7

Simplify.

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Step 1.7.1

Rewrite $\frac{1}{2} (\frac{2}{3} {u_{1}}^{\frac{3}{2}} + C_{1})$ as $\frac{1}{2} \cdot \frac{2}{3} {u_{1}}^{\frac{3}{2}} + C_{1}$ .

$\frac{d y}{d x} = \frac{1}{2} \cdot \frac{2}{3} {u_{1}}^{\frac{3}{2}} + C_{1}$

Step 1.7.2

Simplify.

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Step 1.7.2.1

Multiply $\frac{1}{2}$ by $\frac{2}{3}$ .

$\frac{d y}{d x} = \frac{2}{2 \cdot 3} {u_{1}}^{\frac{3}{2}} + C_{1}$

Step 1.7.2.2

Multiply $2$ by $3$ .

$\frac{d y}{d x} = \frac{2}{6} {u_{1}}^{\frac{3}{2}} + C_{1}$

Step 1.7.2.3

Cancel the common factor of $2$ and $6$ .

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Step 1.7.2.3.1

Factor $2$ out of $2$ .

$\frac{d y}{d x} = \frac{2 (1)}{6} {u_{1}}^{\frac{3}{2}} + C_{1}$

Step 1.7.2.3.2

Cancel the common factors.

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Step 1.7.2.3.2.1

Factor $2$ out of $6$ .

$\frac{d y}{d x} = \frac{2 \cdot 1}{2 \cdot 3} {u_{1}}^{\frac{3}{2}} + C_{1}$

Step 1.7.2.3.2.2

Cancel the common factor.

$\frac{d y}{d x} = \frac{2 \cdot 1}{2 \cdot 3} {u_{1}}^{\frac{3}{2}} + C_{1}$

Step 1.7.2.3.2.3

Rewrite the expression.

$\frac{d y}{d x} = \frac{1}{3} {u_{1}}^{\frac{3}{2}} + C_{1}$

Step 1.8

Replace all occurrences of $u_{1}$ with $2 x - 1$ .

$\frac{d y}{d x} = \frac{1}{3} {(2 x - 1)}^{\frac{3}{2}} + C_{1}$

Step 2

Rewrite the equation.

$d y = (\frac{1}{3} {(2 x - 1)}^{\frac{3}{2}} + C_{1}) d x$

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

$\int d y = \int \frac{1}{3} {(2 x - 1)}^{\frac{3}{2}} + C_{1} d x$

Step 3.2

Apply the constant rule.

$y + C_{2} = \int \frac{1}{3} {(2 x - 1)}^{\frac{3}{2}} + C_{1} d x$

Step 3.3

Integrate the right side.

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Step 3.3.1

Split the single integral into multiple integrals.

$y + C_{2} = \int \frac{1}{3} {(2 x - 1)}^{\frac{3}{2}} d x + \int C_{1} d x$

Step 3.3.2

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

$y + C_{2} = \frac{1}{3} \int {(2 x - 1)}^{\frac{3}{2}} d x + \int C_{1} d x$

Step 3.3.3

Let $u_{2} = 2 x - 1$ . Then $d u_{2} = 2 d x$ , so $\frac{1}{2} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 3.3.3.1

Let $u_{2} = 2 x - 1$ . Find $\frac{d u_{2}}{d x}$ .

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Step 3.3.3.1.1

Differentiate $2 x - 1$ .

$\frac{d}{d x} [2 x - 1]$

Step 3.3.3.1.2

By the Sum Rule, the derivative of $2 x - 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$

Step 3.3.3.1.3

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 3.3.3.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 3.3.3.1.3.2

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

$2 \cdot 1 + \frac{d}{d x} [- 1]$

Step 3.3.3.1.3.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [- 1]$

Step 3.3.3.1.4

Differentiate using the Constant Rule.

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Step 3.3.3.1.4.1

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

$2 + 0$

Step 3.3.3.1.4.2

Add $2$ and $0$ .

$2$

Step 3.3.3.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$y + C_{2} = \frac{1}{3} \int {u_{2}}^{\frac{3}{2}} \frac{1}{2} d u_{2} + \int C_{1} d x$

Step 3.3.4

Combine ${u_{2}}^{\frac{3}{2}}$ and $\frac{1}{2}$ .

$y + C_{2} = \frac{1}{3} \int \frac{{u_{2}}^{\frac{3}{2}}}{2} d u_{2} + \int C_{1} d x$

Step 3.3.5

Since $\frac{1}{2}$ is constant with respect to $u_{2}$ , move $\frac{1}{2}$ out of the integral.

$y + C_{2} = \frac{1}{3} (\frac{1}{2} \int {u_{2}}^{\frac{3}{2}} d u_{2}) + \int C_{1} d x$

Step 3.3.6

Simplify.

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Step 3.3.6.1

Multiply $\frac{1}{2}$ by $\frac{1}{3}$ .

$y + C_{2} = \frac{1}{2 \cdot 3} \int {u_{2}}^{\frac{3}{2}} d u_{2} + \int C_{1} d x$

Step 3.3.6.2

Multiply $2$ by $3$ .

$y + C_{2} = \frac{1}{6} \int {u_{2}}^{\frac{3}{2}} d u_{2} + \int C_{1} d x$

Step 3.3.7

By the Power Rule, the integral of ${u_{2}}^{\frac{3}{2}}$ with respect to $u_{2}$ is $\frac{2}{5} {u_{2}}^{\frac{5}{2}}$ .

$y + C_{2} = \frac{1}{6} (\frac{2}{5} {u_{2}}^{\frac{5}{2}} + C_{3}) + \int C_{1} d x$

Step 3.3.8

Apply the constant rule.

$y + C_{2} = \frac{1}{6} (\frac{2}{5} {u_{2}}^{\frac{5}{2}} + C_{3}) + C_{1} x + C_{4}$

Step 3.3.9

Simplify.

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Step 3.3.9.1

Simplify.

$y + C_{2} = \frac{1}{6} \cdot \frac{2}{5} {u_{2}}^{\frac{5}{2}} + C_{1} x + C_{5}$

Step 3.3.9.2

Simplify.

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Step 3.3.9.2.1

Multiply $\frac{1}{6}$ by $\frac{2}{5}$ .

$y + C_{2} = \frac{2}{6 \cdot 5} {u_{2}}^{\frac{5}{2}} + C_{1} x + C_{5}$

Step 3.3.9.2.2

Multiply $6$ by $5$ .

$y + C_{2} = \frac{2}{30} {u_{2}}^{\frac{5}{2}} + C_{1} x + C_{5}$

Step 3.3.9.2.3

Cancel the common factor of $2$ and $30$ .

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Step 3.3.9.2.3.1

Factor $2$ out of $2$ .

$y + C_{2} = \frac{2 (1)}{30} {u_{2}}^{\frac{5}{2}} + C_{1} x + C_{5}$

Step 3.3.9.2.3.2

Cancel the common factors.

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Step 3.3.9.2.3.2.1

Factor $2$ out of $30$ .

$y + C_{2} = \frac{2 \cdot 1}{2 \cdot 15} {u_{2}}^{\frac{5}{2}} + C_{1} x + C_{5}$

Step 3.3.9.2.3.2.2

Cancel the common factor.

$y + C_{2} = \frac{2 \cdot 1}{2 \cdot 15} {u_{2}}^{\frac{5}{2}} + C_{1} x + C_{5}$

Step 3.3.9.2.3.2.3

Rewrite the expression.

$y + C_{2} = \frac{1}{15} {u_{2}}^{\frac{5}{2}} + C_{1} x + C_{5}$

Step 3.3.10

Replace all occurrences of $u_{2}$ with $2 x - 1$ .

$y + C_{2} = \frac{1}{15} {(2 x - 1)}^{\frac{5}{2}} + C_{1} x + C_{5}$

Step 3.4

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

$y = \frac{1}{15} {(2 x - 1)}^{\frac{5}{2}} + C x + D$