Calculus Examples

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

Step 1

Find the first derivative.

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Find the first derivative.

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Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = x - 3$ .

$f' (x) = x^{2} \frac{d}{d x} (x - 3) + (x - 3) \frac{d}{d x} (x^{2})$

Differentiate.

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

$f' (x) = x^{2} (\frac{d}{d x} (x) + \frac{d}{d x} (- 3)) + (x - 3) \frac{d}{d x} (x^{2})$

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

$f' (x) = x^{2} (1 + \frac{d}{d x} (- 3)) + (x - 3) \frac{d}{d x} (x^{2})$

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$f' (x) = x^{2} (1 + 0) + (x - 3) \frac{d}{d x} (x^{2})$

Simplify the expression.

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Add $1$ and $0$ .

$f' (x) = x^{2} \cdot 1 + (x - 3) \frac{d}{d x} (x^{2})$

Multiply $x^{2}$ by $1$ .

$f' (x) = x^{2} + (x - 3) \frac{d}{d x} (x^{2})$

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

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

Move $2$ to the left of $x - 3$ .

$f' (x) = x^{2} + 2 \cdot ((x - 3) x)$

Simplify.

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Apply the distributive property.

$f' (x) = x^{2} + (2 x + 2 \cdot - 3) x$

Apply the distributive property.

$f' (x) = x^{2} + 2 x \cdot x + 2 \cdot (- 3 x)$

Combine terms.

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Raise $x$ to the power of $1$ .

$f' (x) = x^{2} + 2 (x \cdot x) + 2 \cdot (- 3 x)$

Raise $x$ to the power of $1$ .

$f' (x) = x^{2} + 2 (x \cdot x) + 2 \cdot (- 3 x)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = x^{2} + 2 x^{1 + 1} + 2 \cdot (- 3 x)$

Add $1$ and $1$ .

$f' (x) = x^{2} + 2 x^{2} + 2 \cdot (- 3 x)$

Multiply $2$ by $- 3$ .

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

Add $x^{2}$ and $2 x^{2}$ .

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

The first derivative of $f (x)$ with respect to $x$ is $3 x^{2} - 6 x$ .

$3 x^{2} - 6 x$

Step 2

Set the first derivative equal to $0$ then solve the equation $3 x^{2} - 6 x = 0$ .

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Set the first derivative equal to $0$ .

$3 x^{2} - 6 x = 0$

Factor $3 x$ out of $3 x^{2} - 6 x$ .

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

$3 x (x) - 6 x = 0$

Factor $3 x$ out of $- 6 x$ .

$3 x (x) + 3 x (- 2) = 0$

Factor $3 x$ out of $3 x (x) + 3 x (- 2)$ .

$3 x (x - 2) = 0$

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

$x = 0$

$x - 2 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 2$ equal to $0$ and solve for $x$ .

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Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

The final solution is all the values that make $3 x (x - 2) = 0$ true.

$x = 0, 2$

Step 3

Find the values where the derivative is undefined.

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The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $x^{2} (x - 3)$ at each $x$ value where the derivative is $0$ or undefined.

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Evaluate at $x = 0$ .

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Substitute $0$ for $x$ .

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

Simplify.

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

$0 ((0) - 3)$

Subtract $3$ from $0$ .

$0 \cdot - 3$

Multiply $0$ by $- 3$ .

$0$

Evaluate at $x = 2$ .

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Substitute $2$ for $x$ .

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

Simplify.

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

$4 ((2) - 3)$

Subtract $3$ from $2$ .

$4 \cdot - 1$

Multiply $4$ by $- 1$ .

$- 4$

List all of the points.

$(0, 0), (2, - 4)$

Step 5