Calculus Examples

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

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 $f (x) = x^{2} - 3 x - 4$ and $g (x) = x - 2$ .

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $x^{2} - 3 x - 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 4]$ .

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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

Step 1.1.2.5

Multiply $- 3$ by $1$ .

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

Step 1.1.2.6

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

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

Step 1.1.2.7

Add $2 x - 3$ and $0$ .

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

Step 1.1.2.8

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

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

Step 1.1.2.9

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

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

Step 1.1.2.10

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

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

Step 1.1.2.11

Simplify the expression.

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Step 1.1.2.11.1

Add $1$ and $0$ .

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

Step 1.1.2.11.2

Multiply $- 1$ by $1$ .

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

Step 1.1.3

Simplify.

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Step 1.1.3.1

Apply the distributive property.

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

Step 1.1.3.2

Simplify the numerator.

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Step 1.1.3.2.1

Simplify each term.

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Step 1.1.3.2.1.1

Expand $(x - 2) (2 x - 3)$ using the FOIL Method.

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Step 1.1.3.2.1.1.1

Apply the distributive property.

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

Step 1.1.3.2.1.1.2

Apply the distributive property.

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

Step 1.1.3.2.1.1.3

Apply the distributive property.

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

Step 1.1.3.2.1.2

Simplify and combine like terms.

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Step 1.1.3.2.1.2.1

Simplify each term.

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Step 1.1.3.2.1.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.1.3.2.1.2.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.3.2.1.2.1.2.1

Move $x$ .

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

Step 1.1.3.2.1.2.1.2.2

Multiply $x$ by $x$ .

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

Step 1.1.3.2.1.2.1.3

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

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

Step 1.1.3.2.1.2.1.4

Multiply $2$ by $- 2$ .

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

Step 1.1.3.2.1.2.1.5

Multiply $- 2$ by $- 3$ .

$f' (x) = \frac{2 x^{2} - 3 x - 4 x + 6 - x^{2} - (- 3 x) + 4}{{(x - 2)}^{2}}$

Step 1.1.3.2.1.2.2

Subtract $4 x$ from $- 3 x$ .

$f' (x) = \frac{2 x^{2} - 7 x + 6 - x^{2} - (- 3 x) + 4}{{(x - 2)}^{2}}$

Step 1.1.3.2.1.3

Multiply $- 3$ by $- 1$ .

$f' (x) = \frac{2 x^{2} - 7 x + 6 - x^{2} + 3 x + 4}{{(x - 2)}^{2}}$

Step 1.1.3.2.1.4

Multiply $- 1$ by $- 4$ .

$f' (x) = \frac{2 x^{2} - 7 x + 6 - x^{2} + 3 x + 4}{{(x - 2)}^{2}}$

Step 1.1.3.2.2

Subtract $x^{2}$ from $2 x^{2}$ .

$f' (x) = \frac{x^{2} - 7 x + 6 + 3 x + 4}{{(x - 2)}^{2}}$

Step 1.1.3.2.3

Add $- 7 x$ and $3 x$ .

$f' (x) = \frac{x^{2} - 4 x + 6 + 4}{{(x - 2)}^{2}}$

Step 1.1.3.2.4

Add $6$ and $4$ .

$f' (x) = \frac{x^{2} - 4 x + 10}{{(x - 2)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{x^{2} - 4 x + 10}{{(x - 2)}^{2}}$ .

$\frac{x^{2} - 4 x + 10}{{(x - 2)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{x^{2} - 4 x + 10}{{(x - 2)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{x^{2} - 4 x + 10}{{(x - 2)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$x^{2} - 4 x + 10 = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.3.2

Substitute the values $a = 1$ , $b = - 4$ , and $c = 10$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot 10)}}{2 \cdot 1}$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Simplify the numerator.

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Step 2.3.3.1.1

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 10}}{2 \cdot 1}$

Step 2.3.3.1.2

Multiply $- 4 \cdot 1 \cdot 10$ .

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Step 2.3.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 10}}{2 \cdot 1}$

Step 2.3.3.1.2.2

Multiply $- 4$ by $10$ .

$x = \frac{4 \pm \sqrt{16 - 40}}{2 \cdot 1}$

Step 2.3.3.1.3

Subtract $40$ from $16$ .

$x = \frac{4 \pm \sqrt{- 24}}{2 \cdot 1}$

Step 2.3.3.1.4

Rewrite $- 24$ as $- 1 (24)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 24}}{2 \cdot 1}$

Step 2.3.3.1.5

Rewrite $\sqrt{- 1 (24)}$ as $\sqrt{- 1} \cdot \sqrt{24}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{24}}{2 \cdot 1}$

Step 2.3.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{4 \pm i \cdot \sqrt{24}}{2 \cdot 1}$

Step 2.3.3.1.7

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 2.3.3.1.7.1

Factor $4$ out of $24$ .

$x = \frac{4 \pm i \cdot \sqrt{4 (6)}}{2 \cdot 1}$

Step 2.3.3.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{4 \pm i \cdot \sqrt{2^{2} \cdot 6}}{2 \cdot 1}$

Step 2.3.3.1.8

Pull terms out from under the radical.

$x = \frac{4 \pm i \cdot (2 \sqrt{6})}{2 \cdot 1}$

Step 2.3.3.1.9

Move $2$ to the left of $i$ .

$x = \frac{4 \pm 2 i \sqrt{6}}{2 \cdot 1}$

Step 2.3.3.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 i \sqrt{6}}{2}$

Step 2.3.3.3

Simplify $\frac{4 \pm 2 i \sqrt{6}}{2}$ .

$x = 2 \pm i \sqrt{6}$

Step 2.3.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.3.4.1

Simplify the numerator.

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Step 2.3.4.1.1

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 10}}{2 \cdot 1}$

Step 2.3.4.1.2

Multiply $- 4 \cdot 1 \cdot 10$ .

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Step 2.3.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 10}}{2 \cdot 1}$

Step 2.3.4.1.2.2

Multiply $- 4$ by $10$ .

$x = \frac{4 \pm \sqrt{16 - 40}}{2 \cdot 1}$

Step 2.3.4.1.3

Subtract $40$ from $16$ .

$x = \frac{4 \pm \sqrt{- 24}}{2 \cdot 1}$

Step 2.3.4.1.4

Rewrite $- 24$ as $- 1 (24)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 24}}{2 \cdot 1}$

Step 2.3.4.1.5

Rewrite $\sqrt{- 1 (24)}$ as $\sqrt{- 1} \cdot \sqrt{24}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{24}}{2 \cdot 1}$

Step 2.3.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{4 \pm i \cdot \sqrt{24}}{2 \cdot 1}$

Step 2.3.4.1.7

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 2.3.4.1.7.1

Factor $4$ out of $24$ .

$x = \frac{4 \pm i \cdot \sqrt{4 (6)}}{2 \cdot 1}$

Step 2.3.4.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{4 \pm i \cdot \sqrt{2^{2} \cdot 6}}{2 \cdot 1}$

Step 2.3.4.1.8

Pull terms out from under the radical.

$x = \frac{4 \pm i \cdot (2 \sqrt{6})}{2 \cdot 1}$

Step 2.3.4.1.9

Move $2$ to the left of $i$ .

$x = \frac{4 \pm 2 i \sqrt{6}}{2 \cdot 1}$

Step 2.3.4.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 i \sqrt{6}}{2}$

Step 2.3.4.3

Simplify $\frac{4 \pm 2 i \sqrt{6}}{2}$ .

$x = 2 \pm i \sqrt{6}$

Step 2.3.4.4

Change the $\pm$ to $+$ .

$x = 2 + i \sqrt{6}$

Step 2.3.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.3.5.1

Simplify the numerator.

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Step 2.3.5.1.1

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 10}}{2 \cdot 1}$

Step 2.3.5.1.2

Multiply $- 4 \cdot 1 \cdot 10$ .

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Step 2.3.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 10}}{2 \cdot 1}$

Step 2.3.5.1.2.2

Multiply $- 4$ by $10$ .

$x = \frac{4 \pm \sqrt{16 - 40}}{2 \cdot 1}$

Step 2.3.5.1.3

Subtract $40$ from $16$ .

$x = \frac{4 \pm \sqrt{- 24}}{2 \cdot 1}$

Step 2.3.5.1.4

Rewrite $- 24$ as $- 1 (24)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 24}}{2 \cdot 1}$

Step 2.3.5.1.5

Rewrite $\sqrt{- 1 (24)}$ as $\sqrt{- 1} \cdot \sqrt{24}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{24}}{2 \cdot 1}$

Step 2.3.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{4 \pm i \cdot \sqrt{24}}{2 \cdot 1}$

Step 2.3.5.1.7

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 2.3.5.1.7.1

Factor $4$ out of $24$ .

$x = \frac{4 \pm i \cdot \sqrt{4 (6)}}{2 \cdot 1}$

Step 2.3.5.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{4 \pm i \cdot \sqrt{2^{2} \cdot 6}}{2 \cdot 1}$

Step 2.3.5.1.8

Pull terms out from under the radical.

$x = \frac{4 \pm i \cdot (2 \sqrt{6})}{2 \cdot 1}$

Step 2.3.5.1.9

Move $2$ to the left of $i$ .

$x = \frac{4 \pm 2 i \sqrt{6}}{2 \cdot 1}$

Step 2.3.5.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 i \sqrt{6}}{2}$

Step 2.3.5.3

Simplify $\frac{4 \pm 2 i \sqrt{6}}{2}$ .

$x = 2 \pm i \sqrt{6}$

Step 2.3.5.4

Change the $\pm$ to $-$ .

$x = 2 - i \sqrt{6}$

Step 2.3.6

The final answer is the combination of both solutions.

$x = 2 + i \sqrt{6}, 2 - i \sqrt{6}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{x^{2} - 4 x + 10}{{(x - 2)}^{2}}$ equal to $0$ to find where the expression is undefined.

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

Step 3.2

Solve for $x$ .

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Step 3.2.1

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

$x - 2 = 0$

Step 3.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 4

Evaluate $\frac{x^{2} - 3 x - 4}{x - 2}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 2$ .

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Step 4.1.1

Substitute $2$ for $x$ .

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

Step 4.1.2

Simplify.

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Step 4.1.2.1

Subtract $2$ from $2$ .

$\frac{{(2)}^{2} - 3 \cdot 2 - 4}{0}$

Step 4.1.2.2

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

Undefined

Step 5

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found