Calculus Examples

$y = 7 x - x^{2}$

Step 1

Reorder $7 x$ and $- x^{2}$ .

$y = - x^{2} + 7 x$

Step 2

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

$f (x) = - x^{2} + 7 x$

Step 3

Find the derivative.

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Step 3.1

By the Sum Rule, the derivative of $- x^{2} + 7 x$ with respect to $x$ is $\frac{d}{d x} [- x^{2}] + \frac{d}{d x} [7 x]$ .

$\frac{d}{d x} [- x^{2}] + \frac{d}{d x} [7 x]$

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

$- \frac{d}{d x} [x^{2}] + \frac{d}{d x} [7 x]$

Step 3.2.2

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

$- (2 x) + \frac{d}{d x} [7 x]$

Step 3.2.3

Multiply $2$ by $- 1$ .

$- 2 x + \frac{d}{d x} [7 x]$

Step 3.3

Evaluate $\frac{d}{d x} [7 x]$ .

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Step 3.3.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$- 2 x + 7 \frac{d}{d x} [x]$

Step 3.3.2

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

$- 2 x + 7 \cdot 1$

Step 3.3.3

Multiply $7$ by $1$ .

$- 2 x + 7$

Step 4

Set the derivative equal to $0$ then solve the equation $- 2 x + 7 = 0$ .

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Step 4.1

Subtract $7$ from both sides of the equation.

$- 2 x = - 7$

Step 4.2

Divide each term in $- 2 x = - 7$ by $- 2$ and simplify.

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Step 4.2.1

Divide each term in $- 2 x = - 7$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 7}{- 2}$

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.

$\frac{- 2 x}{- 2} = \frac{- 7}{- 2}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 7}{- 2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{7}{2}$

Step 5

Solve the original function $f (x) = - x^{2} + 7 x$ at $x = \frac{7}{2}$ .

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Step 5.1

Replace the variable $x$ with $\frac{7}{2}$ in the expression.

$f (\frac{7}{2}) = - {(\frac{7}{2})}^{2} + 7 (\frac{7}{2})$

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

Apply the product rule to $\frac{7}{2}$ .

$f (\frac{7}{2}) = - \frac{7^{2}}{2^{2}} + 7 (\frac{7}{2})$

Step 5.2.1.2

Raise $7$ to the power of $2$ .

$f (\frac{7}{2}) = - \frac{49}{2^{2}} + 7 (\frac{7}{2})$

Step 5.2.1.3

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

$f (\frac{7}{2}) = - \frac{49}{4} + 7 (\frac{7}{2})$

Step 5.2.1.4

Multiply $7 (\frac{7}{2})$ .

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Step 5.2.1.4.1

Combine $7$ and $\frac{7}{2}$ .

$f (\frac{7}{2}) = - \frac{49}{4} + \frac{7 \cdot 7}{2}$

Step 5.2.1.4.2

Multiply $7$ by $7$ .

$f (\frac{7}{2}) = - \frac{49}{4} + \frac{49}{2}$

Step 5.2.2

To write $\frac{49}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{7}{2}) = - \frac{49}{4} + \frac{49}{2} \cdot \frac{2}{2}$

Step 5.2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.3.1

Multiply $\frac{49}{2}$ by $\frac{2}{2}$ .

$f (\frac{7}{2}) = - \frac{49}{4} + \frac{49 \cdot 2}{2 \cdot 2}$

Step 5.2.3.2

Multiply $2$ by $2$ .

$f (\frac{7}{2}) = - \frac{49}{4} + \frac{49 \cdot 2}{4}$

Step 5.2.4

Combine the numerators over the common denominator.

$f (\frac{7}{2}) = \frac{- 49 + 49 \cdot 2}{4}$

Step 5.2.5

Simplify the numerator.

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Step 5.2.5.1

Multiply $49$ by $2$ .

$f (\frac{7}{2}) = \frac{- 49 + 98}{4}$

Step 5.2.5.2

Add $- 49$ and $98$ .

$f (\frac{7}{2}) = \frac{49}{4}$

Step 5.2.6

The final answer is $\frac{49}{4}$ .

$\frac{49}{4}$

Step 6

The horizontal tangent line on function $f (x) = - x^{2} + 7 x$ is $y = \frac{49}{4}$ .

$y = \frac{49}{4}$

Step 7