Calculus Examples

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

Step 1

Find the first derivative.

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Find the first derivative.

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Differentiate.

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

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

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

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

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

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Since $- \frac{3}{2}$ is constant with respect to $x$ , the derivative of $- \frac{3}{2} x^{2}$ with respect to $x$ is $- \frac{3}{2} \frac{d}{d x} [x^{2}]$ .

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

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

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

Multiply $2$ by $- 1$ .

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

Combine $- 2$ and $\frac{3}{2}$ .

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

Multiply $- 2$ by $3$ .

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

Combine $\frac{- 6}{2}$ and $x$ .

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

Cancel the common factor of $- 6$ and $2$ .

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Factor $2$ out of $- 6 x$ .

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

Cancel the common factors.

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Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 3 x$ by $1$ .

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

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

$3 x^{2} - 3 x$

Step 2

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

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Set the first derivative equal to $0$ .

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

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

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

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

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

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

Factor $3 x$ out of $3 x (x) + 3 x (- 1)$ .

$3 x (x - 1) = 0$

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

$x = 0$

$x - 1 = 0$

Set $x$ equal to $0$ .

$x = 0$

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

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

$x = 0, 1$

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 $x^{3} - \frac{3}{2} x^{2}$ 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 $x$ .

${(0)}^{3} - \frac{3}{2} \cdot {(0)}^{2}$

Simplify.

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Simplify each term.

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

$0 - \frac{3}{2} \cdot {(0)}^{2}$

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

$0 - \frac{3}{2} \cdot 0$

Multiply $- \frac{3}{2} \cdot 0$ .

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Multiply $0$ by $- 1$ .

$0 + 0 (\frac{3}{2})$

Multiply $0$ by $\frac{3}{2}$ .

$0 + 0$

Add $0$ and $0$ .

$0$

Evaluate at $x = 1$ .

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Substitute $1$ for $x$ .

${(1)}^{3} - \frac{3}{2} \cdot {(1)}^{2}$

Simplify.

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Simplify each term.

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One to any power is one.

$1 - \frac{3}{2} \cdot {(1)}^{2}$

One to any power is one.

$1 - \frac{3}{2} \cdot 1$

Multiply $- 1$ by $1$ .

$1 - \frac{3}{2}$

Simplify the expression.

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

$\frac{2}{2} - \frac{3}{2}$

Combine the numerators over the common denominator.

$\frac{2 - 3}{2}$

Subtract $3$ from $2$ .

$\frac{- 1}{2}$

Move the negative in front of the fraction.

$- \frac{1}{2}$

List all of the points.

$(0, 0), (1, - \frac{1}{2})$

Step 5