Calculus Examples

$y = x^{2} - 5 x + 6$

Step 1

Set $y$ as a function of $x$ .

$f (x) = x^{2} - 5 x + 6$

Step 2

Find the derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $x^{2} - 5 x + 6$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [6]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [6]$

Step 2.1.2

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

$2 x + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [6]$

Step 2.2

Evaluate $\frac{d}{d x} [- 5 x]$ .

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Step 2.2.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$2 x - 5 \frac{d}{d x} [x] + \frac{d}{d x} [6]$

Step 2.2.2

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

$2 x - 5 \cdot 1 + \frac{d}{d x} [6]$

Step 2.2.3

Multiply $- 5$ by $1$ .

$2 x - 5 + \frac{d}{d x} [6]$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$2 x - 5 + 0$

Step 2.3.2

Add $2 x - 5$ and $0$ .

$2 x - 5$

Step 3

Set the derivative equal to $0$ then solve the equation $2 x - 5 = 0$ .

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Step 3.1

Add $5$ to both sides of the equation.

$2 x = 5$

Step 3.2

Divide each term in $2 x = 5$ by $2$ and simplify.

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Step 3.2.1

Divide each term in $2 x = 5$ by $2$ .

$\frac{2 x}{2} = \frac{5}{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{5}{2}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{2}$

Step 4

Solve the original function $f (x) = x^{2} - 5 x + 6$ at $x = \frac{5}{2}$ .

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Step 4.1

Replace the variable $x$ with $\frac{5}{2}$ in the expression.

$f (\frac{5}{2}) = {(\frac{5}{2})}^{2} - 5 (\frac{5}{2}) + 6$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Apply the product rule to $\frac{5}{2}$ .

$f (\frac{5}{2}) = \frac{5^{2}}{2^{2}} - 5 (\frac{5}{2}) + 6$

Step 4.2.1.2

Raise $5$ to the power of $2$ .

$f (\frac{5}{2}) = \frac{25}{2^{2}} - 5 (\frac{5}{2}) + 6$

Step 4.2.1.3

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

$f (\frac{5}{2}) = \frac{25}{4} - 5 (\frac{5}{2}) + 6$

Step 4.2.1.4

Multiply $- 5 (\frac{5}{2})$ .

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Step 4.2.1.4.1

Combine $- 5$ and $\frac{5}{2}$ .

$f (\frac{5}{2}) = \frac{25}{4} + \frac{- 5 \cdot 5}{2} + 6$

Step 4.2.1.4.2

Multiply $- 5$ by $5$ .

$f (\frac{5}{2}) = \frac{25}{4} + \frac{- 25}{2} + 6$

Step 4.2.1.5

Move the negative in front of the fraction.

$f (\frac{5}{2}) = \frac{25}{4} - \frac{25}{2} + 6$

Step 4.2.2

Find the common denominator.

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Step 4.2.2.1

Multiply $\frac{25}{2}$ by $\frac{2}{2}$ .

$f (\frac{5}{2}) = \frac{25}{4} - (\frac{25}{2} \cdot \frac{2}{2}) + 6$

Step 4.2.2.2

Multiply $\frac{25}{2}$ by $\frac{2}{2}$ .

$f (\frac{5}{2}) = \frac{25}{4} - \frac{25 \cdot 2}{2 \cdot 2} + 6$

Step 4.2.2.3

Write $6$ as a fraction with denominator $1$ .

$f (\frac{5}{2}) = \frac{25}{4} - \frac{25 \cdot 2}{2 \cdot 2} + \frac{6}{1}$

Step 4.2.2.4

Multiply $\frac{6}{1}$ by $\frac{4}{4}$ .

$f (\frac{5}{2}) = \frac{25}{4} - \frac{25 \cdot 2}{2 \cdot 2} + \frac{6}{1} \cdot \frac{4}{4}$

Step 4.2.2.5

Multiply $\frac{6}{1}$ by $\frac{4}{4}$ .

$f (\frac{5}{2}) = \frac{25}{4} - \frac{25 \cdot 2}{2 \cdot 2} + \frac{6 \cdot 4}{4}$

Step 4.2.2.6

Multiply $2$ by $2$ .

$f (\frac{5}{2}) = \frac{25}{4} - \frac{25 \cdot 2}{4} + \frac{6 \cdot 4}{4}$

Step 4.2.3

Combine the numerators over the common denominator.

$f (\frac{5}{2}) = \frac{25 - 25 \cdot 2 + 6 \cdot 4}{4}$

Step 4.2.4

Simplify each term.

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Step 4.2.4.1

Multiply $- 25$ by $2$ .

$f (\frac{5}{2}) = \frac{25 - 50 + 6 \cdot 4}{4}$

Step 4.2.4.2

Multiply $6$ by $4$ .

$f (\frac{5}{2}) = \frac{25 - 50 + 24}{4}$

Step 4.2.5

Simplify the expression.

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Step 4.2.5.1

Subtract $50$ from $25$ .

$f (\frac{5}{2}) = \frac{- 25 + 24}{4}$

Step 4.2.5.2

Add $- 25$ and $24$ .

$f (\frac{5}{2}) = \frac{- 1}{4}$

Step 4.2.5.3

Move the negative in front of the fraction.

$f (\frac{5}{2}) = - \frac{1}{4}$

Step 4.2.6

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

$- \frac{1}{4}$

Step 5

The horizontal tangent line on function $f (x) = x^{2} - 5 x + 6$ is $y = - \frac{1}{4}$ .

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

Step 6

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