Calculus Examples

$y = x^{3} - 14 x^{2} + 9 x$

Step 1

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

$f (x) = x^{3} - 14 x^{2} + 9 x$

Step 2

Find the derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $x^{3} - 14 x^{2} + 9 x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 14 x^{2}] + \frac{d}{d x} [9 x]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 14 x^{2}] + \frac{d}{d x} [9 x]$

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [- 14 x^{2}] + \frac{d}{d x} [9 x]$

Step 2.2

Evaluate $\frac{d}{d x} [- 14 x^{2}]$ .

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Step 2.2.1

Since $- 14$ is constant with respect to $x$ , the derivative of $- 14 x^{2}$ with respect to $x$ is $- 14 \frac{d}{d x} [x^{2}]$ .

$3 x^{2} - 14 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [9 x]$

Step 2.2.2

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

$3 x^{2} - 14 (2 x) + \frac{d}{d x} [9 x]$

Step 2.2.3

Multiply $2$ by $- 14$ .

$3 x^{2} - 28 x + \frac{d}{d x} [9 x]$

Step 2.3

Evaluate $\frac{d}{d x} [9 x]$ .

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Step 2.3.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$3 x^{2} - 28 x + 9 \frac{d}{d x} [x]$

Step 2.3.2

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

$3 x^{2} - 28 x + 9 \cdot 1$

Step 2.3.3

Multiply $9$ by $1$ .

$3 x^{2} - 28 x + 9$

Step 3

Set the derivative equal to $0$ then solve the equation $3 x^{2} - 28 x + 9 = 0$ .

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Step 3.1

Factor by grouping.

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Step 3.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 9 = 27$ and whose sum is $b = - 28$ .

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Step 3.1.1.1

Factor $- 28$ out of $- 28 x$ .

$3 x^{2} - 28 x + 9 = 0$

Step 3.1.1.2

Rewrite $- 28$ as $- 1$ plus $- 27$

$3 x^{2} + (- 1 - 27) x + 9 = 0$

Step 3.1.1.3

Apply the distributive property.

$3 x^{2} - 1 x - 27 x + 9 = 0$

Step 3.1.2

Factor out the greatest common factor from each group.

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Step 3.1.2.1

Group the first two terms and the last two terms.

$(3 x^{2} - 1 x) - 27 x + 9 = 0$

Step 3.1.2.2

Factor out the greatest common factor (GCF) from each group.

$x (3 x - 1) - 9 (3 x - 1) = 0$

Step 3.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$(3 x - 1) (x - 9) = 0$

Step 3.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$3 x - 1 = 0$

$x - 9 = 0$

Step 3.3

Set $3 x - 1$ equal to $0$ and solve for $x$ .

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Step 3.3.1

Set $3 x - 1$ equal to $0$ .

$3 x - 1 = 0$

Step 3.3.2

Solve $3 x - 1 = 0$ for $x$ .

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Step 3.3.2.1

Add $1$ to both sides of the equation.

$3 x = 1$

Step 3.3.2.2

Divide each term in $3 x = 1$ by $3$ and simplify.

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Step 3.3.2.2.1

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

Step 3.3.2.2.2

Simplify the left side.

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Step 3.3.2.2.2.1

Cancel the common factor of $3$ .

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Step 3.3.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Step 3.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{3}$

Step 3.4

Set $x - 9$ equal to $0$ and solve for $x$ .

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Step 3.4.1

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 3.4.2

Add $9$ to both sides of the equation.

$x = 9$

Step 3.5

The final solution is all the values that make $(3 x - 1) (x - 9) = 0$ true.

$x = \frac{1}{3}, 9$

Step 4

Solve the original function $f (x) = x^{3} - 14 x^{2} + 9 x$ at $x = \frac{1}{3}$ .

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Step 4.1

Replace the variable $x$ with $\frac{1}{3}$ in the expression.

$f (\frac{1}{3}) = {(\frac{1}{3})}^{3} - 14 {(\frac{1}{3})}^{2} + 9 (\frac{1}{3})$

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

Apply the product rule to $\frac{1}{3}$ .

$f (\frac{1}{3}) = \frac{1^{3}}{3^{3}} - 14 {(\frac{1}{3})}^{2} + 9 (\frac{1}{3})$

Step 4.2.1.2

One to any power is one.

$f (\frac{1}{3}) = \frac{1}{3^{3}} - 14 {(\frac{1}{3})}^{2} + 9 (\frac{1}{3})$

Step 4.2.1.3

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

$f (\frac{1}{3}) = \frac{1}{27} - 14 {(\frac{1}{3})}^{2} + 9 (\frac{1}{3})$

Step 4.2.1.4

Apply the product rule to $\frac{1}{3}$ .

$f (\frac{1}{3}) = \frac{1}{27} - 14 \frac{1^{2}}{3^{2}} + 9 (\frac{1}{3})$

Step 4.2.1.5

One to any power is one.

$f (\frac{1}{3}) = \frac{1}{27} - 14 \frac{1}{3^{2}} + 9 (\frac{1}{3})$

Step 4.2.1.6

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

$f (\frac{1}{3}) = \frac{1}{27} - 14 (\frac{1}{9}) + 9 (\frac{1}{3})$

Step 4.2.1.7

Combine $- 14$ and $\frac{1}{9}$ .

$f (\frac{1}{3}) = \frac{1}{27} + \frac{- 14}{9} + 9 (\frac{1}{3})$

Step 4.2.1.8

Move the negative in front of the fraction.

$f (\frac{1}{3}) = \frac{1}{27} - \frac{14}{9} + 9 (\frac{1}{3})$

Step 4.2.1.9

Cancel the common factor of $3$ .

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Step 4.2.1.9.1

Factor $3$ out of $9$ .

$f (\frac{1}{3}) = \frac{1}{27} - \frac{14}{9} + 3 (3) (\frac{1}{3})$

Step 4.2.1.9.2

Cancel the common factor.

$f (\frac{1}{3}) = \frac{1}{27} - \frac{14}{9} + 3 \cdot (3 (\frac{1}{3}))$

Step 4.2.1.9.3

Rewrite the expression.

$f (\frac{1}{3}) = \frac{1}{27} - \frac{14}{9} + 3$

Step 4.2.2

Find the common denominator.

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Step 4.2.2.1

Multiply $\frac{14}{9}$ by $\frac{3}{3}$ .

$f (\frac{1}{3}) = \frac{1}{27} - (\frac{14}{9} \cdot \frac{3}{3}) + 3$

Step 4.2.2.2

Multiply $\frac{14}{9}$ by $\frac{3}{3}$ .

$f (\frac{1}{3}) = \frac{1}{27} - \frac{14 \cdot 3}{9 \cdot 3} + 3$

Step 4.2.2.3

Write $3$ as a fraction with denominator $1$ .

$f (\frac{1}{3}) = \frac{1}{27} - \frac{14 \cdot 3}{9 \cdot 3} + \frac{3}{1}$

Step 4.2.2.4

Multiply $\frac{3}{1}$ by $\frac{27}{27}$ .

$f (\frac{1}{3}) = \frac{1}{27} - \frac{14 \cdot 3}{9 \cdot 3} + \frac{3}{1} \cdot \frac{27}{27}$

Step 4.2.2.5

Multiply $\frac{3}{1}$ by $\frac{27}{27}$ .

$f (\frac{1}{3}) = \frac{1}{27} - \frac{14 \cdot 3}{9 \cdot 3} + \frac{3 \cdot 27}{27}$

Step 4.2.2.6

Reorder the factors of $9 \cdot 3$ .

$f (\frac{1}{3}) = \frac{1}{27} - \frac{14 \cdot 3}{3 \cdot 9} + \frac{3 \cdot 27}{27}$

Step 4.2.2.7

Multiply $3$ by $9$ .

$f (\frac{1}{3}) = \frac{1}{27} - \frac{14 \cdot 3}{27} + \frac{3 \cdot 27}{27}$

Step 4.2.3

Combine the numerators over the common denominator.

$f (\frac{1}{3}) = \frac{1 - 14 \cdot 3 + 3 \cdot 27}{27}$

Step 4.2.4

Simplify each term.

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Step 4.2.4.1

Multiply $- 14$ by $3$ .

$f (\frac{1}{3}) = \frac{1 - 42 + 3 \cdot 27}{27}$

Step 4.2.4.2

Multiply $3$ by $27$ .

$f (\frac{1}{3}) = \frac{1 - 42 + 81}{27}$

Step 4.2.5

Simplify by adding and subtracting.

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Step 4.2.5.1

Subtract $42$ from $1$ .

$f (\frac{1}{3}) = \frac{- 41 + 81}{27}$

Step 4.2.5.2

Add $- 41$ and $81$ .

$f (\frac{1}{3}) = \frac{40}{27}$

Step 4.2.6

The final answer is $\frac{40}{27}$ .

$\frac{40}{27}$

Step 5

Solve the original function $f (x) = x^{3} - 14 x^{2} + 9 x$ at $x = 9$ .

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Step 5.1

Replace the variable $x$ with $9$ in the expression.

$f (9) = {(9)}^{3} - 14 {(9)}^{2} + 9 (9)$

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 $9$ to the power of $3$ .

$f (9) = 729 - 14 {(9)}^{2} + 9 (9)$

Step 5.2.1.2

Raise $9$ to the power of $2$ .

$f (9) = 729 - 14 \cdot 81 + 9 (9)$

Step 5.2.1.3

Multiply $- 14$ by $81$ .

$f (9) = 729 - 1134 + 9 (9)$

Step 5.2.1.4

Multiply $9$ by $9$ .

$f (9) = 729 - 1134 + 81$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $1134$ from $729$ .

$f (9) = - 405 + 81$

Step 5.2.2.2

Add $- 405$ and $81$ .

$f (9) = - 324$

Step 5.2.3

The final answer is $- 324$ .

$- 324$

Step 6

The horizontal tangent lines on function $f (x) = x^{3} - 14 x^{2} + 9 x$ are $y = \frac{40}{27}, y = - 324$ .

$y = \frac{40}{27}, y = - 324$

Step 7