Calculus Examples

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

Step 1

Find the derivative.

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Step 1.1

Differentiate.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

$3 x^{2} - 2 (2 x)$

Step 1.2.3

Multiply $2$ by $- 2$ .

$3 x^{2} - 4 x$

Step 2

Set the derivative equal to $0$ then solve the equation $3 x^{2} - 4 x = 0$ .

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Step 2.1

Factor $x$ out of $3 x^{2} - 4 x$ .

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Step 2.1.1

Factor $x$ out of $3 x^{2}$ .

$x (3 x) - 4 x = 0$

Step 2.1.2

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

$x (3 x) + x \cdot - 4 = 0$

Step 2.1.3

Factor $x$ out of $x (3 x) + x \cdot - 4$ .

$x (3 x - 4) = 0$

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 $0$ .

$x = 0$

$3 x - 4 = 0$

Step 2.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.4

Set $3 x - 4$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $3 x - 4$ equal to $0$ .

$3 x - 4 = 0$

Step 2.4.2

Solve $3 x - 4 = 0$ for $x$ .

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Step 2.4.2.1

Add $4$ to both sides of the equation.

$3 x = 4$

Step 2.4.2.2

Divide each term in $3 x = 4$ by $3$ and simplify.

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Step 2.4.2.2.1

Divide each term in $3 x = 4$ by $3$ .

$\frac{3 x}{3} = \frac{4}{3}$

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 $3$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{4}{3}$

Step 2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{3}$

Step 2.5

The final solution is all the values that make $x (3 x - 4) = 0$ true.

$x = 0, \frac{4}{3}$

Step 3

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

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Step 3.1

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

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

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 $0$ .

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

Step 3.2.1.2

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

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

Step 3.2.1.3

Multiply $- 2$ by $0$ .

$f (0) = 0 + 0$

Step 3.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 3.2.3

The final answer is $0$ .

$0$

Step 4

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

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Step 4.1

Replace the variable $x$ with $\frac{4}{3}$ in the expression.

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

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 $\frac{4}{3}$ .

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

Step 4.2.1.2

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

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

Step 4.2.1.3

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

$f (\frac{4}{3}) = \frac{64}{27} - 2 {(\frac{4}{3})}^{2}$

Step 4.2.1.4

Apply the product rule to $\frac{4}{3}$ .

$f (\frac{4}{3}) = \frac{64}{27} - 2 \frac{4^{2}}{3^{2}}$

Step 4.2.1.5

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

$f (\frac{4}{3}) = \frac{64}{27} - 2 \frac{16}{3^{2}}$

Step 4.2.1.6

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

$f (\frac{4}{3}) = \frac{64}{27} - 2 (\frac{16}{9})$

Step 4.2.1.7

Multiply $- 2 (\frac{16}{9})$ .

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Step 4.2.1.7.1

Combine $- 2$ and $\frac{16}{9}$ .

$f (\frac{4}{3}) = \frac{64}{27} + \frac{- 2 \cdot 16}{9}$

Step 4.2.1.7.2

Multiply $- 2$ by $16$ .

$f (\frac{4}{3}) = \frac{64}{27} + \frac{- 32}{9}$

Step 4.2.1.8

Move the negative in front of the fraction.

$f (\frac{4}{3}) = \frac{64}{27} - \frac{32}{9}$

Step 4.2.2

To write $- \frac{32}{9}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{4}{3}) = \frac{64}{27} - \frac{32}{9} \cdot \frac{3}{3}$

Step 4.2.3

Write each expression with a common denominator of $27$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.3.1

Multiply $\frac{32}{9}$ by $\frac{3}{3}$ .

$f (\frac{4}{3}) = \frac{64}{27} - \frac{32 \cdot 3}{9 \cdot 3}$

Step 4.2.3.2

Multiply $9$ by $3$ .

$f (\frac{4}{3}) = \frac{64}{27} - \frac{32 \cdot 3}{27}$

Step 4.2.4

Combine the numerators over the common denominator.

$f (\frac{4}{3}) = \frac{64 - 32 \cdot 3}{27}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $- 32$ by $3$ .

$f (\frac{4}{3}) = \frac{64 - 96}{27}$

Step 4.2.5.2

Subtract $96$ from $64$ .

$f (\frac{4}{3}) = \frac{- 32}{27}$

Step 4.2.6

Move the negative in front of the fraction.

$f (\frac{4}{3}) = - \frac{32}{27}$

Step 4.2.7

The final answer is $- \frac{32}{27}$ .

$- \frac{32}{27}$

Step 5

The horizontal tangent lines on function $f (x) = x^{3} - 2 x^{2}$ are $y = 0, y = - \frac{32}{27}$ .

$y = 0, y = - \frac{32}{27}$

Step 6