Calculus Examples

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

Step 1

Find the derivative.

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Step 1.1

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

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

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

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $2$ by $4$ .

$8 x + \frac{d}{d x} [5 (x)]$

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

$8 x + 5 \cdot 1$

Step 1.3.3

Multiply $5$ by $1$ .

$8 x + 5$

Step 2

Set the derivative equal to $0$ then solve the equation $8 x + 5 = 0$ .

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Step 2.1

Subtract $5$ from both sides of the equation.

$8 x = - 5$

Step 2.2

Divide each term in $8 x = - 5$ by $8$ and simplify.

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Step 2.2.1

Divide each term in $8 x = - 5$ by $8$ .

$\frac{8 x}{8} = \frac{- 5}{8}$

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.

$\frac{8 x}{8} = \frac{- 5}{8}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{8}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{5}{8}$

Step 3

Solve the original function $f (x) = 4 {(x)}^{2} + 5 (x)$ at $x = - \frac{5}{8}$ .

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Step 3.1

Replace the variable $x$ with $- \frac{5}{8}$ in the expression.

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

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.2.1.1.1

Apply the product rule to $- \frac{5}{8}$ .

$f (- \frac{5}{8}) = 4 ({(- 1)}^{2} {(\frac{5}{8})}^{2}) + 5 (- \frac{5}{8})$

Step 3.2.1.1.2

Apply the product rule to $\frac{5}{8}$ .

$f (- \frac{5}{8}) = 4 ({(- 1)}^{2} (\frac{5^{2}}{8^{2}})) + 5 (- \frac{5}{8})$

Step 3.2.1.2

Raise $- 1$ to the power of $2$ .

$f (- \frac{5}{8}) = 4 (1 (\frac{5^{2}}{8^{2}})) + 5 (- \frac{5}{8})$

Step 3.2.1.3

Multiply $\frac{5^{2}}{8^{2}}$ by $1$ .

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

Step 3.2.1.4

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

$f (- \frac{5}{8}) = 4 (\frac{25}{8^{2}}) + 5 (- \frac{5}{8})$

Step 3.2.1.5

Raise $8$ to the power of $2$ .

$f (- \frac{5}{8}) = 4 (\frac{25}{64}) + 5 (- \frac{5}{8})$

Step 3.2.1.6

Cancel the common factor of $4$ .

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Step 3.2.1.6.1

Factor $4$ out of $64$ .

$f (- \frac{5}{8}) = 4 (\frac{25}{4 (16)}) + 5 (- \frac{5}{8})$

Step 3.2.1.6.2

Cancel the common factor.

$f (- \frac{5}{8}) = 4 (\frac{25}{4 \cdot 16}) + 5 (- \frac{5}{8})$

Step 3.2.1.6.3

Rewrite the expression.

$f (- \frac{5}{8}) = \frac{25}{16} + 5 (- \frac{5}{8})$

Step 3.2.1.7

Multiply $5 (- \frac{5}{8})$ .

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Step 3.2.1.7.1

Multiply $- 1$ by $5$ .

$f (- \frac{5}{8}) = \frac{25}{16} - 5 (\frac{5}{8})$

Step 3.2.1.7.2

Combine $- 5$ and $\frac{5}{8}$ .

$f (- \frac{5}{8}) = \frac{25}{16} + \frac{- 5 \cdot 5}{8}$

Step 3.2.1.7.3

Multiply $- 5$ by $5$ .

$f (- \frac{5}{8}) = \frac{25}{16} + \frac{- 25}{8}$

Step 3.2.1.8

Move the negative in front of the fraction.

$f (- \frac{5}{8}) = \frac{25}{16} - \frac{25}{8}$

Step 3.2.2

To write $- \frac{25}{8}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- \frac{5}{8}) = \frac{25}{16} - \frac{25}{8} \cdot \frac{2}{2}$

Step 3.2.3

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.2.3.1

Multiply $\frac{25}{8}$ by $\frac{2}{2}$ .

$f (- \frac{5}{8}) = \frac{25}{16} - \frac{25 \cdot 2}{8 \cdot 2}$

Step 3.2.3.2

Multiply $8$ by $2$ .

$f (- \frac{5}{8}) = \frac{25}{16} - \frac{25 \cdot 2}{16}$

Step 3.2.4

Combine the numerators over the common denominator.

$f (- \frac{5}{8}) = \frac{25 - 25 \cdot 2}{16}$

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $- 25$ by $2$ .

$f (- \frac{5}{8}) = \frac{25 - 50}{16}$

Step 3.2.5.2

Subtract $50$ from $25$ .

$f (- \frac{5}{8}) = \frac{- 25}{16}$

Step 3.2.6

Move the negative in front of the fraction.

$f (- \frac{5}{8}) = - \frac{25}{16}$

Step 3.2.7

The final answer is $- \frac{25}{16}$ .

$- \frac{25}{16}$

Step 4

The horizontal tangent line on function $f (x) = 4 {(x)}^{2} + 5 (x)$ is $y = - \frac{25}{16}$ .

$y = - \frac{25}{16}$

Step 5