Calculus Examples

$\frac{x^{2} + 2}{2 x - 1}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{2} + 2$ and $g (x) = 2 x - 1$ .

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

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

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 1.1.2.3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{(2 x - 1) (2 x + 0) - (x^{2} + 2) \frac{d}{d x} [2 x - 1]}{{(2 x - 1)}^{2}}$

Step 1.1.2.4

Simplify the expression.

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Step 1.1.2.4.1

Add $2 x$ and $0$ .

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

Step 1.1.2.4.2

Move $2$ to the left of $2 x - 1$ .

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

Step 1.1.2.5

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

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

Step 1.1.2.6

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

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

Step 1.1.2.7

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

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

Step 1.1.2.8

Multiply $2$ by $1$ .

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

Step 1.1.2.9

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

$\frac{2 (2 x - 1) x - (x^{2} + 2) (2 + 0)}{{(2 x - 1)}^{2}}$

Step 1.1.2.10

Simplify the expression.

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Step 1.1.2.10.1

Add $2$ and $0$ .

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

Step 1.1.2.10.2

Multiply $2$ by $- 1$ .

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

Step 1.1.3

Simplify.

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Step 1.1.3.1

Apply the distributive property.

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

Step 1.1.3.2

Apply the distributive property.

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

Step 1.1.3.3

Apply the distributive property.

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

Step 1.1.3.4

Simplify the numerator.

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Step 1.1.3.4.1

Simplify each term.

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Step 1.1.3.4.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.3.4.1.1.1

Move $x$ .

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

Step 1.1.3.4.1.1.2

Multiply $x$ by $x$ .

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

Step 1.1.3.4.1.2

Multiply $2$ by $2$ .

$\frac{4 x^{2} + 2 \cdot - 1 x - 2 x^{2} - 2 \cdot 2}{{(2 x - 1)}^{2}}$

Step 1.1.3.4.1.3

Multiply $2$ by $- 1$ .

$\frac{4 x^{2} - 2 x - 2 x^{2} - 2 \cdot 2}{{(2 x - 1)}^{2}}$

Step 1.1.3.4.1.4

Multiply $- 2$ by $2$ .

$\frac{4 x^{2} - 2 x - 2 x^{2} - 4}{{(2 x - 1)}^{2}}$

Step 1.1.3.4.2

Subtract $2 x^{2}$ from $4 x^{2}$ .

$\frac{2 x^{2} - 2 x - 4}{{(2 x - 1)}^{2}}$

Step 1.1.3.5

Simplify the numerator.

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Step 1.1.3.5.1

Factor $2$ out of $2 x^{2} - 2 x - 4$ .

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Step 1.1.3.5.1.1

Factor $2$ out of $2 x^{2}$ .

$\frac{2 (x^{2}) - 2 x - 4}{{(2 x - 1)}^{2}}$

Step 1.1.3.5.1.2

Factor $2$ out of $- 2 x$ .

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

Step 1.1.3.5.1.3

Factor $2$ out of $- 4$ .

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

Step 1.1.3.5.1.4

Factor $2$ out of $2 (x^{2}) + 2 (- x)$ .

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

Step 1.1.3.5.1.5

Factor $2$ out of $2 (x^{2} - x) + 2 (- 2)$ .

$\frac{2 (x^{2} - x - 2)}{{(2 x - 1)}^{2}}$

Step 1.1.3.5.2

Factor $x^{2} - x - 2$ using the AC method.

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Step 1.1.3.5.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 1.1.3.5.2.2

Write the factored form using these integers.

$f' (x) = \frac{2 ((x - 2) (x + 1))}{{(2 x - 1)}^{2}}$

$f' (x) = \frac{2 (x - 2) (x + 1)}{{(2 x - 1)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{2 (x - 2) (x + 1)}{{(2 x - 1)}^{2}}$ .

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

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{2 (x - 2) (x + 1)}{{(2 x - 1)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{2 (x - 2) (x + 1)}{{(2 x - 1)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$2 (x - 2) (x + 1) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 2 = 0$

$x + 1 = 0$

Step 2.3.2

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 2.3.2.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 2.3.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.3.3

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 2.3.3.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 2.3.3.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.3.4

The final solution is all the values that make $2 (x - 2) (x + 1) = 0$ true.

$x = 2, - 1$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{2 (x - 2) (x + 1)}{{(2 x - 1)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(2 x - 1)}^{2} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Set the $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Step 3.2.2

Solve for $x$ .

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Step 3.2.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 3.2.2.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 3.2.2.2.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 3.2.2.2.2

Simplify the left side.

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Step 3.2.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 3.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 4

Evaluate $\frac{x^{2} + 2}{2 x - 1}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 2$ .

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Step 4.1.1

Substitute $2$ for $x$ .

$\frac{{(2)}^{2} + 2}{2 (2) - 1}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify the numerator.

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Step 4.1.2.1.1

Raise $2$ to the power of $2$ .

$\frac{4 + 2}{2 (2) - 1}$

Step 4.1.2.1.2

Add $4$ and $2$ .

$\frac{6}{2 (2) - 1}$

Step 4.1.2.2

Simplify the denominator.

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Step 4.1.2.2.1

Multiply $2$ by $2$ .

$\frac{6}{4 - 1}$

Step 4.1.2.2.2

Subtract $1$ from $4$ .

$\frac{6}{3}$

Step 4.1.2.3

Divide $6$ by $3$ .

$2$

Step 4.2

Evaluate at $x = - 1$ .

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Step 4.2.1

Substitute $- 1$ for $x$ .

$\frac{{(- 1)}^{2} + 2}{2 (- 1) - 1}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify the numerator.

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Step 4.2.2.1.1

Raise $- 1$ to the power of $2$ .

$\frac{1 + 2}{2 (- 1) - 1}$

Step 4.2.2.1.2

Add $1$ and $2$ .

$\frac{3}{2 (- 1) - 1}$

Step 4.2.2.2

Simplify the denominator.

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Step 4.2.2.2.1

Multiply $2$ by $- 1$ .

$\frac{3}{- 2 - 1}$

Step 4.2.2.2.2

Subtract $1$ from $- 2$ .

$\frac{3}{- 3}$

Step 4.2.2.3

Divide $3$ by $- 3$ .

$- 1$

Step 4.3

Evaluate at $x = \frac{1}{2}$ .

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Step 4.3.1

Substitute $\frac{1}{2}$ for $x$ .

$\frac{{(\frac{1}{2})}^{2} + 2}{2 (\frac{1}{2}) - 1}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Cancel the common factor of $2$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{{(\frac{1}{2})}^{2} + 2}{2 (\frac{1}{2}) - 1}$

Step 4.3.2.1.2

Rewrite the expression.

$\frac{{(\frac{1}{2})}^{2} + 2}{1 - 1}$

Step 4.3.2.2

Subtract $1$ from $1$ .

$\frac{{(\frac{1}{2})}^{2} + 2}{0}$

Step 4.3.2.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 4.4

List all of the points.

$(2, 2), (- 1, - 1)$

Step 5