Calculus Examples

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

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 $f (x) = x^{2} - 2 x + 1$ and $g (x) = x$ .

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

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

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 $n = 2$ .

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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

Step 1.1.2.5

Multiply $- 2$ by $1$ .

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

Step 1.1.2.6

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

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

Step 1.1.2.7

Add $2 x - 2$ and $0$ .

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

Step 1.1.2.8

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

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

Step 1.1.2.9

Multiply $- 1$ by $1$ .

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

Step 1.1.3

Simplify.

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Step 1.1.3.1

Apply the distributive property.

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

Step 1.1.3.2

Apply the distributive property.

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

Step 1.1.3.3

Simplify the numerator.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.1.3.3.1.2

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

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Step 1.1.3.3.1.2.1

Move $x$ .

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

Step 1.1.3.3.1.2.2

Multiply $x$ by $x$ .

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

Step 1.1.3.3.1.3

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

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

Step 1.1.3.3.1.4

Multiply $- 2$ by $- 1$ .

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

Step 1.1.3.3.1.5

Multiply $- 1$ by $1$ .

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

Step 1.1.3.3.2

Combine the opposite terms in $2 x^{2} - 2 x - x^{2} + 2 x - 1$ .

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Step 1.1.3.3.2.1

Add $- 2 x$ and $2 x$ .

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

Step 1.1.3.3.2.2

Add $2 x^{2} - x^{2}$ and $0$ .

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

Step 1.1.3.3.3

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

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

Step 1.1.3.4

Simplify the numerator.

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Step 1.1.3.4.1

Rewrite $1$ as $1^{2}$ .

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

Step 1.1.3.4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

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

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 $0$ .

$x + 1 = 0$

$x - 1 = 0$

Step 2.3.2

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

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Step 2.3.2.1

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

$x + 1 = 0$

Step 2.3.2.2

Subtract $1$ from both sides of the equation.

$x = - 1$

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

Add $1$ to both sides of the equation.

$x = 1$

Step 2.3.4

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

$x = - 1, 1$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

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

$x^{2} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 3.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 3.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 4

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

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Step 4.1

Evaluate at $x = - 1$ .

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Step 4.1.1

Substitute $- 1$ for $x$ .

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

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify the expression.

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Step 4.1.2.1.1

Move the negative one from the denominator of $\frac{{(- 1)}^{2} - 2 \cdot - 1 + 1}{- 1}$ .

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

Step 4.1.2.1.2

Rewrite $- 1 \cdot ({(- 1)}^{2} - 2 \cdot - 1 + 1)$ as $- ({(- 1)}^{2} - 2 \cdot - 1 + 1)$ .

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

Step 4.1.2.2

Simplify each term.

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Step 4.1.2.2.1

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

$- (1 - 2 \cdot - 1 + 1)$

Step 4.1.2.2.2

Multiply $- 2$ by $- 1$ .

$- (1 + 2 + 1)$

Step 4.1.2.3

Simplify the expression.

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Step 4.1.2.3.1

Add $1$ and $2$ .

$- (3 + 1)$

Step 4.1.2.3.2

Add $3$ and $1$ .

$- 1 \cdot 4$

Step 4.1.2.3.3

Multiply $- 1$ by $4$ .

$- 4$

Step 4.2

Evaluate at $x = 1$ .

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Step 4.2.1

Substitute $1$ for $x$ .

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

Divide ${(1)}^{2} - 2 \cdot 1 + 1$ by $1$ .

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

Step 4.2.2.2

Simplify each term.

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Step 4.2.2.2.1

One to any power is one.

$1 - 2 \cdot 1 + 1$

Step 4.2.2.2.2

Multiply $- 2$ by $1$ .

$1 - 2 + 1$

Step 4.2.2.3

Simplify by adding and subtracting.

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Step 4.2.2.3.1

Subtract $2$ from $1$ .

$- 1 + 1$

Step 4.2.2.3.2

Add $- 1$ and $1$ .

$0$

Step 4.3

Evaluate at $x = 0$ .

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Step 4.3.1

Substitute $0$ for $x$ .

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

Step 4.3.2

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

Undefined

Step 4.4

List all of the points.

$(- 1, - 4), (1, 0)$

Step 5