Calculus Examples

$x^{2} - x$

Step 1

Find the derivative.

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Step 1.1

Differentiate.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.2

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

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Step 1.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

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

Step 1.2.2

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

$2 x - 1 \cdot 1$

Step 1.2.3

Multiply $- 1$ by $1$ .

$2 x - 1$

Step 2

Set the derivative equal to $0$ then solve the equation $2 x - 1 = 0$ .

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Step 2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.1.2

Divide $x$ by $1$ .

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

Step 3

Solve the original function $f (x) = x^{2} - x$ at $x = \frac{1}{2}$ .

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Step 3.1

Replace the variable $x$ with $\frac{1}{2}$ in the expression.

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

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Apply the product rule to $\frac{1}{2}$ .

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

Step 3.2.1.2

One to any power is one.

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

Step 3.2.1.3

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

$f (\frac{1}{2}) = \frac{1}{4} - \frac{1}{2}$

Step 3.2.2

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{1}{2}) = \frac{1}{4} - \frac{1}{2} \cdot \frac{2}{2}$

Step 3.2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.2.3.1

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

$f (\frac{1}{2}) = \frac{1}{4} - \frac{2}{2 \cdot 2}$

Step 3.2.3.2

Multiply $2$ by $2$ .

$f (\frac{1}{2}) = \frac{1}{4} - \frac{2}{4}$

Step 3.2.4

Combine the numerators over the common denominator.

$f (\frac{1}{2}) = \frac{1 - 2}{4}$

Step 3.2.5

Subtract $2$ from $1$ .

$f (\frac{1}{2}) = \frac{- 1}{4}$

Step 3.2.6

Move the negative in front of the fraction.

$f (\frac{1}{2}) = - \frac{1}{4}$

Step 3.2.7

The final answer is $- \frac{1}{4}$ .

$- \frac{1}{4}$

Step 4

The horizontal tangent line on function $f (x) = x^{2} - x$ is $y = - \frac{1}{4}$ .

$y = - \frac{1}{4}$

Step 5