Calculus Examples

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

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 $x^{2} - 3 x$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x]$ .

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

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

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

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

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

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

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

$f' (x) = 2 x - 3 \cdot 1$

Multiply $- 3$ by $1$ .

$f' (x) = 2 x - 3$

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

$2 x - 3$

Step 2

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

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Set the first derivative equal to $0$ .

$2 x - 3 = 0$

Add $3$ to both sides of the equation.

$2 x = 3$

Divide each term in $2 x = 3$ by $2$ and simplify.

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Divide each term in $2 x = 3$ by $2$ .

$\frac{2 x}{2} = \frac{3}{2}$

Simplify the left side.

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Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 x}{2} = \frac{3}{2}$

Divide $x$ by $1$ .

$x = \frac{3}{2}$

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

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Evaluate at $x = \frac{3}{2}$ .

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Substitute $\frac{3}{2}$ for $x$ .

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

Simplify.

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Simplify each term.

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Apply the product rule to $\frac{3}{2}$ .

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

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

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

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

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

Multiply $- 3 (\frac{3}{2})$ .

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

$\frac{9}{4} + \frac{- 3 \cdot 3}{2}$

Multiply $- 3$ by $3$ .

$\frac{9}{4} + \frac{- 9}{2}$

Move the negative in front of the fraction.

$\frac{9}{4} - \frac{9}{2}$

To write $- \frac{9}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{9}{4} - \frac{9}{2} \cdot \frac{2}{2}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{9}{2}$ by $\frac{2}{2}$ .

$\frac{9}{4} - \frac{9 \cdot 2}{2 \cdot 2}$

Multiply $2$ by $2$ .

$\frac{9}{4} - \frac{9 \cdot 2}{4}$

Combine the numerators over the common denominator.

$\frac{9 - 9 \cdot 2}{4}$

Simplify the numerator.

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Multiply $- 9$ by $2$ .

$\frac{9 - 18}{4}$

Subtract $18$ from $9$ .

$\frac{- 9}{4}$

Move the negative in front of the fraction.

$- \frac{9}{4}$

List all of the points.

$(\frac{3}{2}, - \frac{9}{4})$

Step 5