Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 1.1.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.1.2.1.2

Multiply $2$ by $2$ .

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

Step 1.1.2.2

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

$f' (x) = \frac{x^{2} (\frac{d}{d x} (x) + \frac{d}{d x} (- 3)) - (x - 3) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.1.2.3

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

$f' (x) = \frac{x^{2} (1 + \frac{d}{d x} (- 3)) - (x - 3) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.1.2.4

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

$f' (x) = \frac{x^{2} (1 + 0) - (x - 3) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.1.2.5

Simplify the expression.

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Step 1.1.2.5.1

Add $1$ and $0$ .

$f' (x) = \frac{x^{2} \cdot 1 - (x - 3) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.1.2.5.2

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

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

Step 1.1.2.6

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

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

Step 1.1.2.7

Simplify with factoring out.

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Step 1.1.2.7.1

Multiply $2$ by $- 1$ .

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

Step 1.1.2.7.2

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

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Step 1.1.2.7.2.1

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

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

Step 1.1.2.7.2.2

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

$f' (x) = \frac{x \cdot x + x (- 2 (x - 3))}{x^{4}}$

Step 1.1.2.7.2.3

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

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

Step 1.1.3

Cancel the common factors.

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Step 1.1.3.1

Factor $x$ out of $x^{4}$ .

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

Step 1.1.3.2

Cancel the common factor.

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

Step 1.1.3.3

Rewrite the expression.

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

Step 1.1.4

Simplify.

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Step 1.1.4.1

Apply the distributive property.

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

Step 1.1.4.2

Simplify the numerator.

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Step 1.1.4.2.1

Multiply $- 2$ by $- 3$ .

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

Step 1.1.4.2.2

Subtract $2 x$ from $x$ .

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

Step 1.1.4.3

Factor $- 1$ out of $- x$ .

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

Step 1.1.4.4

Rewrite $6$ as $- 1 (- 6)$ .

$f' (x) = \frac{- (x) - 1 \cdot - 6}{x^{3}}$

Step 1.1.4.5

Factor $- 1$ out of $- (x) - 1 (- 6)$ .

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

Step 1.1.4.6

Rewrite $- (x - 6)$ as $- 1 (x - 6)$ .

$f' (x) = \frac{- 1 (x - 6)}{x^{3}}$

Step 1.1.4.7

Move the negative in front of the fraction.

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

Step 1.2

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

$- \frac{x - 6}{x^{3}}$

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{x - 6}{x^{3}} = 0$

Step 2.2

Set the numerator equal to zero.

$x - 6 = 0$

Step 2.3

Add $6$ to both sides of the equation.

$x = 6$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{x - 6}{x^{3}}$ equal to $0$ to find where the expression is undefined.

$x^{3} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{0}$

Step 3.2.2

Simplify $\sqrt[3]{0}$ .

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Step 3.2.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 3.2.2.2

Pull terms out from under the radical, assuming real numbers.

$x = 0$

Step 4

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

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Step 4.1

Evaluate at $x = 6$ .

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Step 4.1.1

Substitute $6$ for $x$ .

$\frac{(6) - 3}{{(6)}^{2}}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify the expression.

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Step 4.1.2.1.1

Subtract $3$ from $6$ .

$\frac{3}{6^{2}}$

Step 4.1.2.1.2

Raise $6$ to the power of $2$ .

$\frac{3}{36}$

Step 4.1.2.2

Cancel the common factor of $3$ and $36$ .

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Step 4.1.2.2.1

Factor $3$ out of $3$ .

$\frac{3 (1)}{36}$

Step 4.1.2.2.2

Cancel the common factors.

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Step 4.1.2.2.2.1

Factor $3$ out of $36$ .

$\frac{3 \cdot 1}{3 \cdot 12}$

Step 4.1.2.2.2.2

Cancel the common factor.

$\frac{3 \cdot 1}{3 \cdot 12}$

Step 4.1.2.2.2.3

Rewrite the expression.

$\frac{1}{12}$

Step 4.2

Evaluate at $x = 0$ .

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Step 4.2.1

Substitute $0$ for $x$ .

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

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

$\frac{(0) - 3}{0}$

Step 4.2.2.2

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 4.3

List all of the points.

$(6, \frac{1}{12})$

Step 5