Calculus Examples

$f (x) = 8 x - 2 x^{2}$

Step 1

Find the derivative.

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Step 1.1

By the Sum Rule, the derivative of $8 x - 2 x^{2}$ with respect to $x$ is $\frac{d}{d x} [8 x] + \frac{d}{d x} [- 2 x^{2}]$ .

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

Step 1.2

Evaluate $\frac{d}{d x} [8 x]$ .

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Step 1.2.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $8$ by $1$ .

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

$8 - 2 (2 x)$

Step 1.3.3

Multiply $2$ by $- 2$ .

$8 - 4 x$

Step 1.4

Reorder terms.

$- 4 x + 8$

Step 2

Set the derivative equal to $0$ then solve the equation $- 4 x + 8 = 0$ .

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Step 2.1

Subtract $8$ from both sides of the equation.

$- 4 x = - 8$

Step 2.2

Divide each term in $- 4 x = - 8$ by $- 4$ and simplify.

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Step 2.2.1

Divide each term in $- 4 x = - 8$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 8}{- 4}$

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.

$\frac{- 4 x}{- 4} = \frac{- 8}{- 4}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 8}{- 4}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 8$ by $- 4$ .

$x = 2$

Step 3

Solve the original function $f (x) = 8 x - 2 x^{2}$ at $x = 2$ .

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Step 3.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = 8 (2) - 2 {(2)}^{2}$

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

Multiply $8$ by $2$ .

$f (2) = 16 - 2 {(2)}^{2}$

Step 3.2.1.2

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

$f (2) = 16 - 2 \cdot 4$

Step 3.2.1.3

Multiply $- 2$ by $4$ .

$f (2) = 16 - 8$

Step 3.2.2

Subtract $8$ from $16$ .

$f (2) = 8$

Step 3.2.3

The final answer is $8$ .

$8$

Step 4

The horizontal tangent line on function $f (x) = 8 x - 2 x^{2}$ is $y = 8$ .

$y = 8$

Step 5