Calculus Examples

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

Step 1

Find the derivative.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $3$ by $3$ .

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Multiply $2$ by $- 2$ .

$9 x^{2} - 4 x + \frac{d}{d x} [- 9]$

Step 1.4

Differentiate using the Constant Rule.

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Step 1.4.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$9 x^{2} - 4 x + 0$

Step 1.4.2

Add $9 x^{2} - 4 x$ and $0$ .

$9 x^{2} - 4 x$

Step 2

Set the derivative equal to $0$ then solve the equation $9 x^{2} - 4 x = 0$ .

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Step 2.1

Factor $x$ out of $9 x^{2} - 4 x$ .

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Step 2.1.1

Factor $x$ out of $9 x^{2}$ .

$x (9 x) - 4 x = 0$

Step 2.1.2

Factor $x$ out of $- 4 x$ .

$x (9 x) + x \cdot - 4 = 0$

Step 2.1.3

Factor $x$ out of $x (9 x) + x \cdot - 4$ .

$x (9 x - 4) = 0$

Step 2.2

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

$x = 0$

$9 x - 4 = 0$

Step 2.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.4

Set $9 x - 4$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $9 x - 4$ equal to $0$ .

$9 x - 4 = 0$

Step 2.4.2

Solve $9 x - 4 = 0$ for $x$ .

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Step 2.4.2.1

Add $4$ to both sides of the equation.

$9 x = 4$

Step 2.4.2.2

Divide each term in $9 x = 4$ by $9$ and simplify.

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Step 2.4.2.2.1

Divide each term in $9 x = 4$ by $9$ .

$\frac{9 x}{9} = \frac{4}{9}$

Step 2.4.2.2.2

Simplify the left side.

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Step 2.4.2.2.2.1

Cancel the common factor of $9$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{4}{9}$

Step 2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{9}$

Step 2.5

The final solution is all the values that make $x (9 x - 4) = 0$ true.

$x = 0, \frac{4}{9}$

Step 3

Solve the original function $f (x) = 3 x^{3} - 2 x^{2} - 9$ at $x = 0$ .

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Step 3.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = 3 {(0)}^{3} - 2 {(0)}^{2} - 9$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Raising $0$ to any positive power yields $0$ .

$f (0) = 3 \cdot 0 - 2 {(0)}^{2} - 9$

Step 3.2.1.2

Multiply $3$ by $0$ .

$f (0) = 0 - 2 {(0)}^{2} - 9$

Step 3.2.1.3

Raising $0$ to any positive power yields $0$ .

$f (0) = 0 - 2 \cdot 0 - 9$

Step 3.2.1.4

Multiply $- 2$ by $0$ .

$f (0) = 0 + 0 - 9$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Add $0$ and $0$ .

$f (0) = 0 - 9$

Step 3.2.2.2

Subtract $9$ from $0$ .

$f (0) = - 9$

Step 3.2.3

The final answer is $- 9$ .

$- 9$

Step 4

Solve the original function $f (x) = 3 x^{3} - 2 x^{2} - 9$ at $x = \frac{4}{9}$ .

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Step 4.1

Replace the variable $x$ with $\frac{4}{9}$ in the expression.

$f (\frac{4}{9}) = 3 {(\frac{4}{9})}^{3} - 2 {(\frac{4}{9})}^{2} - 9$

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 $\frac{4}{9}$ .

$f (\frac{4}{9}) = 3 (\frac{4^{3}}{9^{3}}) - 2 {(\frac{4}{9})}^{2} - 9$

Step 4.2.1.2

Raise $4$ to the power of $3$ .

$f (\frac{4}{9}) = 3 (\frac{64}{9^{3}}) - 2 {(\frac{4}{9})}^{2} - 9$

Step 4.2.1.3

Raise $9$ to the power of $3$ .

$f (\frac{4}{9}) = 3 (\frac{64}{729}) - 2 {(\frac{4}{9})}^{2} - 9$

Step 4.2.1.4

Cancel the common factor of $3$ .

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Step 4.2.1.4.1

Factor $3$ out of $729$ .

$f (\frac{4}{9}) = 3 (\frac{64}{3 (243)}) - 2 {(\frac{4}{9})}^{2} - 9$

Step 4.2.1.4.2

Cancel the common factor.

$f (\frac{4}{9}) = 3 (\frac{64}{3 \cdot 243}) - 2 {(\frac{4}{9})}^{2} - 9$

Step 4.2.1.4.3

Rewrite the expression.

$f (\frac{4}{9}) = \frac{64}{243} - 2 {(\frac{4}{9})}^{2} - 9$

Step 4.2.1.5

Apply the product rule to $\frac{4}{9}$ .

$f (\frac{4}{9}) = \frac{64}{243} - 2 \frac{4^{2}}{9^{2}} - 9$

Step 4.2.1.6

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

$f (\frac{4}{9}) = \frac{64}{243} - 2 \frac{16}{9^{2}} - 9$

Step 4.2.1.7

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

$f (\frac{4}{9}) = \frac{64}{243} - 2 (\frac{16}{81}) - 9$

Step 4.2.1.8

Multiply $- 2 (\frac{16}{81})$ .

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Step 4.2.1.8.1

Combine $- 2$ and $\frac{16}{81}$ .

$f (\frac{4}{9}) = \frac{64}{243} + \frac{- 2 \cdot 16}{81} - 9$

Step 4.2.1.8.2

Multiply $- 2$ by $16$ .

$f (\frac{4}{9}) = \frac{64}{243} + \frac{- 32}{81} - 9$

Step 4.2.1.9

Move the negative in front of the fraction.

$f (\frac{4}{9}) = \frac{64}{243} - \frac{32}{81} - 9$

Step 4.2.2

Find the common denominator.

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Step 4.2.2.1

Multiply $\frac{32}{81}$ by $\frac{3}{3}$ .

$f (\frac{4}{9}) = \frac{64}{243} - (\frac{32}{81} \cdot \frac{3}{3}) - 9$

Step 4.2.2.2

Multiply $\frac{32}{81}$ by $\frac{3}{3}$ .

$f (\frac{4}{9}) = \frac{64}{243} - \frac{32 \cdot 3}{81 \cdot 3} - 9$

Step 4.2.2.3

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

$f (\frac{4}{9}) = \frac{64}{243} - \frac{32 \cdot 3}{81 \cdot 3} + \frac{- 9}{1}$

Step 4.2.2.4

Multiply $\frac{- 9}{1}$ by $\frac{243}{243}$ .

$f (\frac{4}{9}) = \frac{64}{243} - \frac{32 \cdot 3}{81 \cdot 3} + \frac{- 9}{1} \cdot \frac{243}{243}$

Step 4.2.2.5

Multiply $\frac{- 9}{1}$ by $\frac{243}{243}$ .

$f (\frac{4}{9}) = \frac{64}{243} - \frac{32 \cdot 3}{81 \cdot 3} + \frac{- 9 \cdot 243}{243}$

Step 4.2.2.6

Reorder the factors of $81 \cdot 3$ .

$f (\frac{4}{9}) = \frac{64}{243} - \frac{32 \cdot 3}{3 \cdot 81} + \frac{- 9 \cdot 243}{243}$

Step 4.2.2.7

Multiply $3$ by $81$ .

$f (\frac{4}{9}) = \frac{64}{243} - \frac{32 \cdot 3}{243} + \frac{- 9 \cdot 243}{243}$

Step 4.2.3

Combine the numerators over the common denominator.

$f (\frac{4}{9}) = \frac{64 - 32 \cdot 3 - 9 \cdot 243}{243}$

Step 4.2.4

Simplify each term.

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Step 4.2.4.1

Multiply $- 32$ by $3$ .

$f (\frac{4}{9}) = \frac{64 - 96 - 9 \cdot 243}{243}$

Step 4.2.4.2

Multiply $- 9$ by $243$ .

$f (\frac{4}{9}) = \frac{64 - 96 - 2187}{243}$

Step 4.2.5

Simplify the expression.

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Step 4.2.5.1

Subtract $96$ from $64$ .

$f (\frac{4}{9}) = \frac{- 32 - 2187}{243}$

Step 4.2.5.2

Subtract $2187$ from $- 32$ .

$f (\frac{4}{9}) = \frac{- 2219}{243}$

Step 4.2.5.3

Move the negative in front of the fraction.

$f (\frac{4}{9}) = - \frac{2219}{243}$

Step 4.2.6

The final answer is $- \frac{2219}{243}$ .

$- \frac{2219}{243}$

Step 5

The horizontal tangent lines on function $f (x) = 3 x^{3} - 2 x^{2} - 9$ are $y = - 9, y = - \frac{2219}{243}$ .

$y = - 9, y = - \frac{2219}{243}$

Step 6