Calculus Examples

$y = 2 + 8 x - x^{2}$

Step 1

Simplify $2 + 8 x - x^{2}$ .

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Step 1.1

Move $2$ .

$y = 8 x - x^{2} + 2$

Step 1.2

Reorder $8 x$ and $- x^{2}$ .

$y = - x^{2} + 8 x + 2$

Step 2

Set $y$ as a function of $x$ .

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

Step 3

Find the derivative.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

Step 3.2.2

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

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

Step 3.2.3

Multiply $2$ by $- 1$ .

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Multiply $8$ by $1$ .

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

Step 3.4

Differentiate using the Constant Rule.

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Step 3.4.1

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$- 2 x + 8 + 0$

Step 3.4.2

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

$- 2 x + 8$

Step 4

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

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Step 4.1

Subtract $8$ from both sides of the equation.

$- 2 x = - 8$

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $- 2$ .

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Step 4.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.1.2

Divide $x$ by $1$ .

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

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $- 8$ by $- 2$ .

$x = 4$

Step 5

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

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Step 5.1

Replace the variable $x$ with $4$ in the expression.

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

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

$f (4) = - 1 \cdot 16 + 8 (4) + 2$

Step 5.2.1.2

Multiply $- 1$ by $16$ .

$f (4) = - 16 + 8 (4) + 2$

Step 5.2.1.3

Multiply $8$ by $4$ .

$f (4) = - 16 + 32 + 2$

Step 5.2.2

Simplify by adding numbers.

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Step 5.2.2.1

Add $- 16$ and $32$ .

$f (4) = 16 + 2$

Step 5.2.2.2

Add $16$ and $2$ .

$f (4) = 18$

Step 5.2.3

The final answer is $18$ .

$18$

Step 6

The horizontal tangent line on function $f (x) = - x^{2} + 8 x + 2$ is $y = 18$ .

$y = 18$

Step 7