Calculus Examples

$f (x) = 5 x^{2} - 4 x + 3$

Step 1

Find the derivative.

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Step 1.1

By the Sum Rule, the derivative of $5 x^{2} - 4 x + 3$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [3]$

Step 1.2

Evaluate $\frac{d}{d x} [5 x^{2}]$ .

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Step 1.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [3]$

Step 1.2.2

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

$5 (2 x) + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [3]$

Step 1.2.3

Multiply $2$ by $5$ .

$10 x + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [3]$

Step 1.3

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

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Step 1.3.1

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

$10 x - 4 \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 1.3.2

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

$10 x - 4 \cdot 1 + \frac{d}{d x} [3]$

Step 1.3.3

Multiply $- 4$ by $1$ .

$10 x - 4 + \frac{d}{d x} [3]$

Step 1.4

Differentiate using the Constant Rule.

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Step 1.4.1

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$10 x - 4 + 0$

Step 1.4.2

Add $10 x - 4$ and $0$ .

$10 x - 4$

Step 2

Set the derivative equal to $0$ then solve the equation $10 x - 4 = 0$ .

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Step 2.1

Add $4$ to both sides of the equation.

$10 x = 4$

Step 2.2

Divide each term in $10 x = 4$ by $10$ and simplify.

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Step 2.2.1

Divide each term in $10 x = 4$ by $10$ .

$\frac{10 x}{10} = \frac{4}{10}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $10$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{10 x}{10} = \frac{4}{10}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{10}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Cancel the common factor of $4$ and $10$ .

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Step 2.2.3.1.1

Factor $2$ out of $4$ .

$x = \frac{2 (2)}{10}$

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 $10$ .

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

Step 2.2.3.1.2.2

Cancel the common factor.

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

Step 2.2.3.1.2.3

Rewrite the expression.

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

Step 3

Solve the original function $f (x) = 5 x^{2} - 4 x + 3$ at $x = \frac{2}{5}$ .

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Step 3.1

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

$f (\frac{2}{5}) = 5 {(\frac{2}{5})}^{2} - 4 (\frac{2}{5}) + 3$

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 $\frac{2}{5}$ .

$f (\frac{2}{5}) = 5 (\frac{2^{2}}{5^{2}}) - 4 (\frac{2}{5}) + 3$

Step 3.2.1.2

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

$f (\frac{2}{5}) = 5 (\frac{4}{5^{2}}) - 4 (\frac{2}{5}) + 3$

Step 3.2.1.3

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

$f (\frac{2}{5}) = 5 (\frac{4}{25}) - 4 (\frac{2}{5}) + 3$

Step 3.2.1.4

Cancel the common factor of $5$ .

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Step 3.2.1.4.1

Factor $5$ out of $25$ .

$f (\frac{2}{5}) = 5 (\frac{4}{5 (5)}) - 4 (\frac{2}{5}) + 3$

Step 3.2.1.4.2

Cancel the common factor.

$f (\frac{2}{5}) = 5 (\frac{4}{5 \cdot 5}) - 4 (\frac{2}{5}) + 3$

Step 3.2.1.4.3

Rewrite the expression.

$f (\frac{2}{5}) = \frac{4}{5} - 4 (\frac{2}{5}) + 3$

Step 3.2.1.5

Multiply $- 4 (\frac{2}{5})$ .

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Step 3.2.1.5.1

Combine $- 4$ and $\frac{2}{5}$ .

$f (\frac{2}{5}) = \frac{4}{5} + \frac{- 4 \cdot 2}{5} + 3$

Step 3.2.1.5.2

Multiply $- 4$ by $2$ .

$f (\frac{2}{5}) = \frac{4}{5} + \frac{- 8}{5} + 3$

Step 3.2.1.6

Move the negative in front of the fraction.

$f (\frac{2}{5}) = \frac{4}{5} - \frac{8}{5} + 3$

Step 3.2.2

Combine fractions.

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Step 3.2.2.1

Combine the numerators over the common denominator.

$f (\frac{2}{5}) = 3 + \frac{4 - 8}{5}$

Step 3.2.2.2

Simplify the expression.

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Step 3.2.2.2.1

Subtract $8$ from $4$ .

$f (\frac{2}{5}) = 3 + \frac{- 4}{5}$

Step 3.2.2.2.2

Move the negative in front of the fraction.

$f (\frac{2}{5}) = 3 - \frac{4}{5}$

Step 3.2.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (\frac{2}{5}) = 3 \cdot \frac{5}{5} - \frac{4}{5}$

Step 3.2.4

Combine $3$ and $\frac{5}{5}$ .

$f (\frac{2}{5}) = \frac{3 \cdot 5}{5} - \frac{4}{5}$

Step 3.2.5

Combine the numerators over the common denominator.

$f (\frac{2}{5}) = \frac{3 \cdot 5 - 4}{5}$

Step 3.2.6

Simplify the numerator.

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Step 3.2.6.1

Multiply $3$ by $5$ .

$f (\frac{2}{5}) = \frac{15 - 4}{5}$

Step 3.2.6.2

Subtract $4$ from $15$ .

$f (\frac{2}{5}) = \frac{11}{5}$

Step 3.2.7

The final answer is $\frac{11}{5}$ .

$\frac{11}{5}$

Step 4

The horizontal tangent line on function $f (x) = 5 x^{2} - 4 x + 3$ is $y = \frac{11}{5}$ .

$y = \frac{11}{5}$

Step 5