Calculus Examples

$y = 2 x^{2} + 6 x - 7$

Step 1

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

$f (x) = 2 x^{2} + 6 x - 7$

Step 2

Find the derivative.

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Step 2.1

By the Sum Rule, the derivative of $2 x^{2} + 6 x - 7$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 7]$ .

$\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 7]$

Step 2.2

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

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Step 2.2.1

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

$2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 7]$

Step 2.2.2

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

$2 (2 x) + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 7]$

Step 2.2.3

Multiply $2$ by $2$ .

$4 x + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 7]$

Step 2.3

Evaluate $\frac{d}{d x} [6 x]$ .

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Step 2.3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$4 x + 6 \frac{d}{d x} [x] + \frac{d}{d x} [- 7]$

Step 2.3.2

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

$4 x + 6 \cdot 1 + \frac{d}{d x} [- 7]$

Step 2.3.3

Multiply $6$ by $1$ .

$4 x + 6 + \frac{d}{d x} [- 7]$

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$4 x + 6 + 0$

Step 2.4.2

Add $4 x + 6$ and $0$ .

$4 x + 6$

Step 3

Set the derivative equal to $0$ then solve the equation $4 x + 6 = 0$ .

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Step 3.1

Subtract $6$ from both sides of the equation.

$4 x = - 6$

Step 3.2

Divide each term in $4 x = - 6$ by $4$ and simplify.

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Step 3.2.1

Divide each term in $4 x = - 6$ by $4$ .

$\frac{4 x}{4} = \frac{- 6}{4}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 6}{4}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 6}{4}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Cancel the common factor of $- 6$ and $4$ .

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Step 3.2.3.1.1

Factor $2$ out of $- 6$ .

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

Step 3.2.3.1.2

Cancel the common factors.

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Step 3.2.3.1.2.1

Factor $2$ out of $4$ .

$x = \frac{2 \cdot - 3}{2 \cdot 2}$

Step 3.2.3.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot - 3}{2 \cdot 2}$

Step 3.2.3.1.2.3

Rewrite the expression.

$x = \frac{- 3}{2}$

Step 3.2.3.2

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

Step 4

Solve the original function $f (x) = 2 x^{2} + 6 x - 7$ at $x = - \frac{3}{2}$ .

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Step 4.1

Replace the variable $x$ with $- \frac{3}{2}$ in the expression.

$f (- \frac{3}{2}) = 2 {(- \frac{3}{2})}^{2} + 6 (- \frac{3}{2}) - 7$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

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

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Step 4.2.1.1.1

Apply the product rule to $- \frac{3}{2}$ .

$f (- \frac{3}{2}) = 2 ({(- 1)}^{2} {(\frac{3}{2})}^{2}) + 6 (- \frac{3}{2}) - 7$

Step 4.2.1.1.2

Apply the product rule to $\frac{3}{2}$ .

$f (- \frac{3}{2}) = 2 ({(- 1)}^{2} (\frac{3^{2}}{2^{2}})) + 6 (- \frac{3}{2}) - 7$

Step 4.2.1.2

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

$f (- \frac{3}{2}) = 2 (1 (\frac{3^{2}}{2^{2}})) + 6 (- \frac{3}{2}) - 7$

Step 4.2.1.3

Multiply $\frac{3^{2}}{2^{2}}$ by $1$ .

$f (- \frac{3}{2}) = 2 (\frac{3^{2}}{2^{2}}) + 6 (- \frac{3}{2}) - 7$

Step 4.2.1.4

Raise $3$ to the power of $2$ .

$f (- \frac{3}{2}) = 2 (\frac{9}{2^{2}}) + 6 (- \frac{3}{2}) - 7$

Step 4.2.1.5

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

$f (- \frac{3}{2}) = 2 (\frac{9}{4}) + 6 (- \frac{3}{2}) - 7$

Step 4.2.1.6

Cancel the common factor of $2$ .

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Step 4.2.1.6.1

Factor $2$ out of $4$ .

$f (- \frac{3}{2}) = 2 (\frac{9}{2 (2)}) + 6 (- \frac{3}{2}) - 7$

Step 4.2.1.6.2

Cancel the common factor.

$f (- \frac{3}{2}) = 2 (\frac{9}{2 \cdot 2}) + 6 (- \frac{3}{2}) - 7$

Step 4.2.1.6.3

Rewrite the expression.

$f (- \frac{3}{2}) = \frac{9}{2} + 6 (- \frac{3}{2}) - 7$

Step 4.2.1.7

Cancel the common factor of $2$ .

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Step 4.2.1.7.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$f (- \frac{3}{2}) = \frac{9}{2} + 6 (\frac{- 3}{2}) - 7$

Step 4.2.1.7.2

Factor $2$ out of $6$ .

$f (- \frac{3}{2}) = \frac{9}{2} + 2 (3) (\frac{- 3}{2}) - 7$

Step 4.2.1.7.3

Cancel the common factor.

$f (- \frac{3}{2}) = \frac{9}{2} + 2 \cdot (3 (\frac{- 3}{2})) - 7$

Step 4.2.1.7.4

Rewrite the expression.

$f (- \frac{3}{2}) = \frac{9}{2} + 3 \cdot - 3 - 7$

Step 4.2.1.8

Multiply $3$ by $- 3$ .

$f (- \frac{3}{2}) = \frac{9}{2} - 9 - 7$

Step 4.2.2

Find the common denominator.

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Step 4.2.2.1

Write $- 9$ as a fraction with denominator $1$ .

$f (- \frac{3}{2}) = \frac{9}{2} + \frac{- 9}{1} - 7$

Step 4.2.2.2

Multiply $\frac{- 9}{1}$ by $\frac{2}{2}$ .

$f (- \frac{3}{2}) = \frac{9}{2} + \frac{- 9}{1} \cdot \frac{2}{2} - 7$

Step 4.2.2.3

Multiply $\frac{- 9}{1}$ by $\frac{2}{2}$ .

$f (- \frac{3}{2}) = \frac{9}{2} + \frac{- 9 \cdot 2}{2} - 7$

Step 4.2.2.4

Write $- 7$ as a fraction with denominator $1$ .

$f (- \frac{3}{2}) = \frac{9}{2} + \frac{- 9 \cdot 2}{2} + \frac{- 7}{1}$

Step 4.2.2.5

Multiply $\frac{- 7}{1}$ by $\frac{2}{2}$ .

$f (- \frac{3}{2}) = \frac{9}{2} + \frac{- 9 \cdot 2}{2} + \frac{- 7}{1} \cdot \frac{2}{2}$

Step 4.2.2.6

Multiply $\frac{- 7}{1}$ by $\frac{2}{2}$ .

$f (- \frac{3}{2}) = \frac{9}{2} + \frac{- 9 \cdot 2}{2} + \frac{- 7 \cdot 2}{2}$

Step 4.2.3

Combine the numerators over the common denominator.

$f (- \frac{3}{2}) = \frac{9 - 9 \cdot 2 - 7 \cdot 2}{2}$

Step 4.2.4

Simplify each term.

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Step 4.2.4.1

Multiply $- 9$ by $2$ .

$f (- \frac{3}{2}) = \frac{9 - 18 - 7 \cdot 2}{2}$

Step 4.2.4.2

Multiply $- 7$ by $2$ .

$f (- \frac{3}{2}) = \frac{9 - 18 - 14}{2}$

Step 4.2.5

Simplify the expression.

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Step 4.2.5.1

Subtract $18$ from $9$ .

$f (- \frac{3}{2}) = \frac{- 9 - 14}{2}$

Step 4.2.5.2

Subtract $14$ from $- 9$ .

$f (- \frac{3}{2}) = \frac{- 23}{2}$

Step 4.2.5.3

Move the negative in front of the fraction.

$f (- \frac{3}{2}) = - \frac{23}{2}$

Step 4.2.6

The final answer is $- \frac{23}{2}$ .

$- \frac{23}{2}$

Step 5

The horizontal tangent line on function $f (x) = 2 x^{2} + 6 x - 7$ is $y = - \frac{23}{2}$ .

$y = - \frac{23}{2}$

Step 6