Calculus Examples

$f (x) = 2 x^{2} - 9 x + 10$

Step 1

Find the derivative.

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Step 1.1

By the Sum Rule, the derivative of $2 x^{2} - 9 x + 10$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 9 x] + \frac{d}{d x} [10]$ .

$\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 9 x] + \frac{d}{d x} [10]$

Step 1.2

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

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Step 1.2.1

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

$2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9 x] + \frac{d}{d x} [10]$

Step 1.2.2

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

$2 (2 x) + \frac{d}{d x} [- 9 x] + \frac{d}{d x} [10]$

Step 1.2.3

Multiply $2$ by $2$ .

$4 x + \frac{d}{d x} [- 9 x] + \frac{d}{d x} [10]$

Step 1.3

Evaluate $\frac{d}{d x} [- 9 x]$ .

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Step 1.3.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x$ with respect to $x$ is $- 9 \frac{d}{d x} [x]$ .

$4 x - 9 \frac{d}{d x} [x] + \frac{d}{d x} [10]$

Step 1.3.2

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

$4 x - 9 \cdot 1 + \frac{d}{d x} [10]$

Step 1.3.3

Multiply $- 9$ by $1$ .

$4 x - 9 + \frac{d}{d x} [10]$

Step 1.4

Differentiate using the Constant Rule.

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Step 1.4.1

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$4 x - 9 + 0$

Step 1.4.2

Add $4 x - 9$ and $0$ .

$4 x - 9$

Step 2

Set the derivative equal to $0$ then solve the equation $4 x - 9 = 0$ .

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Step 2.1

Add $9$ to both sides of the equation.

$4 x = 9$

Step 2.2

Divide each term in $4 x = 9$ by $4$ and simplify.

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Step 2.2.1

Divide each term in $4 x = 9$ by $4$ .

$\frac{4 x}{4} = \frac{9}{4}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $4$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{9}{4}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{9}{4}$

Step 3

Solve the original function $f (x) = 2 x^{2} - 9 x + 10$ at $x = \frac{9}{4}$ .

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Step 3.1

Replace the variable $x$ with $\frac{9}{4}$ in the expression.

$f (\frac{9}{4}) = 2 {(\frac{9}{4})}^{2} - 9 (\frac{9}{4}) + 10$

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 $\frac{9}{4}$ .

$f (\frac{9}{4}) = 2 (\frac{9^{2}}{4^{2}}) - 9 (\frac{9}{4}) + 10$

Step 3.2.1.2

Raise $9$ to the power of $2$ .

$f (\frac{9}{4}) = 2 (\frac{81}{4^{2}}) - 9 (\frac{9}{4}) + 10$

Step 3.2.1.3

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

$f (\frac{9}{4}) = 2 (\frac{81}{16}) - 9 (\frac{9}{4}) + 10$

Step 3.2.1.4

Cancel the common factor of $2$ .

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Step 3.2.1.4.1

Factor $2$ out of $16$ .

$f (\frac{9}{4}) = 2 (\frac{81}{2 (8)}) - 9 (\frac{9}{4}) + 10$

Step 3.2.1.4.2

Cancel the common factor.

$f (\frac{9}{4}) = 2 (\frac{81}{2 \cdot 8}) - 9 (\frac{9}{4}) + 10$

Step 3.2.1.4.3

Rewrite the expression.

$f (\frac{9}{4}) = \frac{81}{8} - 9 (\frac{9}{4}) + 10$

Step 3.2.1.5

Multiply $- 9 (\frac{9}{4})$ .

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Step 3.2.1.5.1

Combine $- 9$ and $\frac{9}{4}$ .

$f (\frac{9}{4}) = \frac{81}{8} + \frac{- 9 \cdot 9}{4} + 10$

Step 3.2.1.5.2

Multiply $- 9$ by $9$ .

$f (\frac{9}{4}) = \frac{81}{8} + \frac{- 81}{4} + 10$

Step 3.2.1.6

Move the negative in front of the fraction.

$f (\frac{9}{4}) = \frac{81}{8} - \frac{81}{4} + 10$

Step 3.2.2

Find the common denominator.

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Step 3.2.2.1

Multiply $\frac{81}{4}$ by $\frac{2}{2}$ .

$f (\frac{9}{4}) = \frac{81}{8} - (\frac{81}{4} \cdot \frac{2}{2}) + 10$

Step 3.2.2.2

Multiply $\frac{81}{4}$ by $\frac{2}{2}$ .

$f (\frac{9}{4}) = \frac{81}{8} - \frac{81 \cdot 2}{4 \cdot 2} + 10$

Step 3.2.2.3

Write $10$ as a fraction with denominator $1$ .

$f (\frac{9}{4}) = \frac{81}{8} - \frac{81 \cdot 2}{4 \cdot 2} + \frac{10}{1}$

Step 3.2.2.4

Multiply $\frac{10}{1}$ by $\frac{8}{8}$ .

$f (\frac{9}{4}) = \frac{81}{8} - \frac{81 \cdot 2}{4 \cdot 2} + \frac{10}{1} \cdot \frac{8}{8}$

Step 3.2.2.5

Multiply $\frac{10}{1}$ by $\frac{8}{8}$ .

$f (\frac{9}{4}) = \frac{81}{8} - \frac{81 \cdot 2}{4 \cdot 2} + \frac{10 \cdot 8}{8}$

Step 3.2.2.6

Reorder the factors of $4 \cdot 2$ .

$f (\frac{9}{4}) = \frac{81}{8} - \frac{81 \cdot 2}{2 \cdot 4} + \frac{10 \cdot 8}{8}$

Step 3.2.2.7

Multiply $2$ by $4$ .

$f (\frac{9}{4}) = \frac{81}{8} - \frac{81 \cdot 2}{8} + \frac{10 \cdot 8}{8}$

Step 3.2.3

Combine the numerators over the common denominator.

$f (\frac{9}{4}) = \frac{81 - 81 \cdot 2 + 10 \cdot 8}{8}$

Step 3.2.4

Simplify each term.

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Step 3.2.4.1

Multiply $- 81$ by $2$ .

$f (\frac{9}{4}) = \frac{81 - 162 + 10 \cdot 8}{8}$

Step 3.2.4.2

Multiply $10$ by $8$ .

$f (\frac{9}{4}) = \frac{81 - 162 + 80}{8}$

Step 3.2.5

Simplify the expression.

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Step 3.2.5.1

Subtract $162$ from $81$ .

$f (\frac{9}{4}) = \frac{- 81 + 80}{8}$

Step 3.2.5.2

Add $- 81$ and $80$ .

$f (\frac{9}{4}) = \frac{- 1}{8}$

Step 3.2.5.3

Move the negative in front of the fraction.

$f (\frac{9}{4}) = - \frac{1}{8}$

Step 3.2.6

The final answer is $- \frac{1}{8}$ .

$- \frac{1}{8}$

Step 4

The horizontal tangent line on function $f (x) = 2 x^{2} - 9 x + 10$ is $y = - \frac{1}{8}$ .

$y = - \frac{1}{8}$

Step 5