Calculus Examples

$f (x) = 4 x^{2} - 2 x + 5$

Step 1

Find the derivative.

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Step 1.1

By the Sum Rule, the derivative of $4 x^{2} - 2 x + 5$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [5]$

Step 1.2

Evaluate $\frac{d}{d x} [4 x^{2}]$ .

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Step 1.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [5]$

Step 1.2.2

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

$4 (2 x) + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [5]$

Step 1.2.3

Multiply $2$ by $4$ .

$8 x + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [5]$

Step 1.3

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

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Step 1.3.1

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

$8 x - 2 \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Step 1.3.2

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

$8 x - 2 \cdot 1 + \frac{d}{d x} [5]$

Step 1.3.3

Multiply $- 2$ by $1$ .

$8 x - 2 + \frac{d}{d x} [5]$

Step 1.4

Differentiate using the Constant Rule.

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Step 1.4.1

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$8 x - 2 + 0$

Step 1.4.2

Add $8 x - 2$ and $0$ .

$8 x - 2$

Step 2

Set the derivative equal to $0$ then solve the equation $8 x - 2 = 0$ .

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Step 2.1

Add $2$ to both sides of the equation.

$8 x = 2$

Step 2.2

Divide each term in $8 x = 2$ by $8$ and simplify.

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Step 2.2.1

Divide each term in $8 x = 2$ by $8$ .

$\frac{8 x}{8} = \frac{2}{8}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $8$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{2}{8}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{8}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Cancel the common factor of $2$ and $8$ .

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Step 2.2.3.1.1

Factor $2$ out of $2$ .

$x = \frac{2 (1)}{8}$

Step 2.2.3.1.2

Cancel the common factors.

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Step 2.2.3.1.2.1

Factor $2$ out of $8$ .

$x = \frac{2 \cdot 1}{2 \cdot 4}$

Step 2.2.3.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 1}{2 \cdot 4}$

Step 2.2.3.1.2.3

Rewrite the expression.

$x = \frac{1}{4}$

Step 3

Solve the original function $f (x) = 4 x^{2} - 2 x + 5$ at $x = \frac{1}{4}$ .

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Step 3.1

Replace the variable $x$ with $\frac{1}{4}$ in the expression.

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

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 $\frac{1}{4}$ .

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

Step 3.2.1.2

One to any power is one.

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

Step 3.2.1.3

Raise $4$ to the power of $2$ .

$f (\frac{1}{4}) = 4 (\frac{1}{16}) - 2 (\frac{1}{4}) + 5$

Step 3.2.1.4

Cancel the common factor of $4$ .

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Step 3.2.1.4.1

Factor $4$ out of $16$ .

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

Step 3.2.1.4.2

Cancel the common factor.

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

Step 3.2.1.4.3

Rewrite the expression.

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

Step 3.2.1.5

Cancel the common factor of $2$ .

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Step 3.2.1.5.1

Factor $2$ out of $- 2$ .

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

Step 3.2.1.5.2

Factor $2$ out of $4$ .

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

Step 3.2.1.5.3

Cancel the common factor.

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

Step 3.2.1.5.4

Rewrite the expression.

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

Step 3.2.1.6

Rewrite $- 1 (\frac{1}{2})$ as $- (\frac{1}{2})$ .

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

Step 3.2.2

Find the common denominator.

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Step 3.2.2.1

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

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

Step 3.2.2.2

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

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

Step 3.2.2.3

Write $5$ as a fraction with denominator $1$ .

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

Step 3.2.2.4

Multiply $\frac{5}{1}$ by $\frac{4}{4}$ .

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

Step 3.2.2.5

Multiply $\frac{5}{1}$ by $\frac{4}{4}$ .

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

Step 3.2.2.6

Multiply $2$ by $2$ .

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

Step 3.2.3

Combine the numerators over the common denominator.

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

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

Multiply $5$ by $4$ .

$f (\frac{1}{4}) = \frac{1 - 2 + 20}{4}$

Step 3.2.4.2

Subtract $2$ from $1$ .

$f (\frac{1}{4}) = \frac{- 1 + 20}{4}$

Step 3.2.4.3

Add $- 1$ and $20$ .

$f (\frac{1}{4}) = \frac{19}{4}$

Step 3.2.5

The final answer is $\frac{19}{4}$ .

$\frac{19}{4}$

Step 4

The horizontal tangent line on function $f (x) = 4 x^{2} - 2 x + 5$ is $y = \frac{19}{4}$ .

$y = \frac{19}{4}$

Step 5