Calculus Examples

$y = 6 x^{2} + 3 x - 6$

Step 1

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

$f (x) = 6 x^{2} + 3 x - 6$

Step 2

Find the derivative.

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Step 2.1

By the Sum Rule, the derivative of $6 x^{2} + 3 x - 6$ with respect to $x$ is $\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 6]$ .

$\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 6]$

Step 2.2

Evaluate $\frac{d}{d x} [6 x^{2}]$ .

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Step 2.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2}$ with respect to $x$ is $6 \frac{d}{d x} [x^{2}]$ .

$6 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 6]$

Step 2.2.2

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

$6 (2 x) + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 6]$

Step 2.2.3

Multiply $2$ by $6$ .

$12 x + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 6]$

Step 2.3

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 2.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$12 x + 3 \frac{d}{d x} [x] + \frac{d}{d x} [- 6]$

Step 2.3.2

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

$12 x + 3 \cdot 1 + \frac{d}{d x} [- 6]$

Step 2.3.3

Multiply $3$ by $1$ .

$12 x + 3 + \frac{d}{d x} [- 6]$

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$12 x + 3 + 0$

Step 2.4.2

Add $12 x + 3$ and $0$ .

$12 x + 3$

Step 3

Set the derivative equal to $0$ then solve the equation $12 x + 3 = 0$ .

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Step 3.1

Subtract $3$ from both sides of the equation.

$12 x = - 3$

Step 3.2

Divide each term in $12 x = - 3$ by $12$ and simplify.

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Step 3.2.1

Divide each term in $12 x = - 3$ by $12$ .

$\frac{12 x}{12} = \frac{- 3}{12}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $12$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{12 x}{12} = \frac{- 3}{12}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{12}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Cancel the common factor of $- 3$ and $12$ .

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Step 3.2.3.1.1

Factor $3$ out of $- 3$ .

$x = \frac{3 (- 1)}{12}$

Step 3.2.3.1.2

Cancel the common factors.

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Step 3.2.3.1.2.1

Factor $3$ out of $12$ .

$x = \frac{3 \cdot - 1}{3 \cdot 4}$

Step 3.2.3.1.2.2

Cancel the common factor.

$x = \frac{3 \cdot - 1}{3 \cdot 4}$

Step 3.2.3.1.2.3

Rewrite the expression.

$x = \frac{- 1}{4}$

Step 3.2.3.2

Move the negative in front of the fraction.

$x = - \frac{1}{4}$

Step 4

Solve the original function $f (x) = 6 x^{2} + 3 x - 6$ at $x = - \frac{1}{4}$ .

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Step 4.1

Replace the variable $x$ with $- \frac{1}{4}$ in the expression.

$f (- \frac{1}{4}) = 6 {(- \frac{1}{4})}^{2} + 3 (- \frac{1}{4}) - 6$

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 $- \frac{1}{4}$ .

$f (- \frac{1}{4}) = 6 ({(- 1)}^{2} {(\frac{1}{4})}^{2}) + 3 (- \frac{1}{4}) - 6$

Step 4.2.1.1.2

Apply the product rule to $\frac{1}{4}$ .

$f (- \frac{1}{4}) = 6 ({(- 1)}^{2} (\frac{1^{2}}{4^{2}})) + 3 (- \frac{1}{4}) - 6$

Step 4.2.1.2

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

$f (- \frac{1}{4}) = 6 (1 (\frac{1^{2}}{4^{2}})) + 3 (- \frac{1}{4}) - 6$

Step 4.2.1.3

Multiply $\frac{1^{2}}{4^{2}}$ by $1$ .

$f (- \frac{1}{4}) = 6 (\frac{1^{2}}{4^{2}}) + 3 (- \frac{1}{4}) - 6$

Step 4.2.1.4

One to any power is one.

$f (- \frac{1}{4}) = 6 (\frac{1}{4^{2}}) + 3 (- \frac{1}{4}) - 6$

Step 4.2.1.5

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

$f (- \frac{1}{4}) = 6 (\frac{1}{16}) + 3 (- \frac{1}{4}) - 6$

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 $6$ .

$f (- \frac{1}{4}) = 2 (3) (\frac{1}{16}) + 3 (- \frac{1}{4}) - 6$

Step 4.2.1.6.2

Factor $2$ out of $16$ .

$f (- \frac{1}{4}) = 2 \cdot (3 (\frac{1}{2 \cdot 8})) + 3 (- \frac{1}{4}) - 6$

Step 4.2.1.6.3

Cancel the common factor.

$f (- \frac{1}{4}) = 2 \cdot (3 (\frac{1}{2 \cdot 8})) + 3 (- \frac{1}{4}) - 6$

Step 4.2.1.6.4

Rewrite the expression.

$f (- \frac{1}{4}) = 3 (\frac{1}{8}) + 3 (- \frac{1}{4}) - 6$

Step 4.2.1.7

Combine $3$ and $\frac{1}{8}$ .

$f (- \frac{1}{4}) = \frac{3}{8} + 3 (- \frac{1}{4}) - 6$

Step 4.2.1.8

Multiply $3 (- \frac{1}{4})$ .

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Step 4.2.1.8.1

Multiply $- 1$ by $3$ .

$f (- \frac{1}{4}) = \frac{3}{8} - 3 (\frac{1}{4}) - 6$

Step 4.2.1.8.2

Combine $- 3$ and $\frac{1}{4}$ .

$f (- \frac{1}{4}) = \frac{3}{8} + \frac{- 3}{4} - 6$

Step 4.2.1.9

Move the negative in front of the fraction.

$f (- \frac{1}{4}) = \frac{3}{8} - \frac{3}{4} - 6$

Step 4.2.2

Find the common denominator.

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Step 4.2.2.1

Multiply $\frac{3}{4}$ by $\frac{2}{2}$ .

$f (- \frac{1}{4}) = \frac{3}{8} - (\frac{3}{4} \cdot \frac{2}{2}) - 6$

Step 4.2.2.2

Multiply $\frac{3}{4}$ by $\frac{2}{2}$ .

$f (- \frac{1}{4}) = \frac{3}{8} - \frac{3 \cdot 2}{4 \cdot 2} - 6$

Step 4.2.2.3

Write $- 6$ as a fraction with denominator $1$ .

$f (- \frac{1}{4}) = \frac{3}{8} - \frac{3 \cdot 2}{4 \cdot 2} + \frac{- 6}{1}$

Step 4.2.2.4

Multiply $\frac{- 6}{1}$ by $\frac{8}{8}$ .

$f (- \frac{1}{4}) = \frac{3}{8} - \frac{3 \cdot 2}{4 \cdot 2} + \frac{- 6}{1} \cdot \frac{8}{8}$

Step 4.2.2.5

Multiply $\frac{- 6}{1}$ by $\frac{8}{8}$ .

$f (- \frac{1}{4}) = \frac{3}{8} - \frac{3 \cdot 2}{4 \cdot 2} + \frac{- 6 \cdot 8}{8}$

Step 4.2.2.6

Reorder the factors of $4 \cdot 2$ .

$f (- \frac{1}{4}) = \frac{3}{8} - \frac{3 \cdot 2}{2 \cdot 4} + \frac{- 6 \cdot 8}{8}$

Step 4.2.2.7

Multiply $2$ by $4$ .

$f (- \frac{1}{4}) = \frac{3}{8} - \frac{3 \cdot 2}{8} + \frac{- 6 \cdot 8}{8}$

Step 4.2.3

Combine the numerators over the common denominator.

$f (- \frac{1}{4}) = \frac{3 - 3 \cdot 2 - 6 \cdot 8}{8}$

Step 4.2.4

Simplify each term.

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Step 4.2.4.1

Multiply $- 3$ by $2$ .

$f (- \frac{1}{4}) = \frac{3 - 6 - 6 \cdot 8}{8}$

Step 4.2.4.2

Multiply $- 6$ by $8$ .

$f (- \frac{1}{4}) = \frac{3 - 6 - 48}{8}$

Step 4.2.5

Simplify the expression.

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Step 4.2.5.1

Subtract $6$ from $3$ .

$f (- \frac{1}{4}) = \frac{- 3 - 48}{8}$

Step 4.2.5.2

Subtract $48$ from $- 3$ .

$f (- \frac{1}{4}) = \frac{- 51}{8}$

Step 4.2.5.3

Move the negative in front of the fraction.

$f (- \frac{1}{4}) = - \frac{51}{8}$

Step 4.2.6

The final answer is $- \frac{51}{8}$ .

$- \frac{51}{8}$

Step 5

The horizontal tangent line on function $f (x) = 6 x^{2} + 3 x - 6$ is $y = - \frac{51}{8}$ .

$y = - \frac{51}{8}$

Step 6