Calculus Examples

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

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 Constant Multiple Rule.

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Step 1.1.1.1

Move the negative in front of the fraction.

$\frac{d}{d x} [- \frac{4}{x^{2} - 2 x - 3}]$

Step 1.1.1.2

Since $- 4$ is constant with respect to $x$ , the derivative of $- \frac{4}{x^{2} - 2 x - 3}$ with respect to $x$ is $- 4 \frac{d}{d x} [\frac{1}{x^{2} - 2 x - 3}]$ .

$- 4 \frac{d}{d x} [\frac{1}{x^{2} - 2 x - 3}]$

Step 1.1.1.3

Rewrite $\frac{1}{x^{2} - 2 x - 3}$ as ${(x^{2} - 2 x - 3)}^{- 1}$ .

$- 4 \frac{d}{d x} [{(x^{2} - 2 x - 3)}^{- 1}]$

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2} - 2 x - 3$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u$ as $x^{2} - 2 x - 3$ .

$- 4 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2} - 2 x - 3])$

Step 1.1.2.2

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

$- 4 (- u^{- 2} \frac{d}{d x} [x^{2} - 2 x - 3])$

Step 1.1.2.3

Replace all occurrences of $u$ with $x^{2} - 2 x - 3$ .

$- 4 (- {(x^{2} - 2 x - 3)}^{- 2} \frac{d}{d x} [x^{2} - 2 x - 3])$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

Multiply $- 1$ by $- 4$ .

$4 ({(x^{2} - 2 x - 3)}^{- 2} \frac{d}{d x} [x^{2} - 2 x - 3])$

Step 1.1.3.2

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

$4 {(x^{2} - 2 x - 3)}^{- 2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 3])$

Step 1.1.3.3

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

$4 {(x^{2} - 2 x - 3)}^{- 2} (2 x + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 3])$

Step 1.1.3.4

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

$4 {(x^{2} - 2 x - 3)}^{- 2} (2 x - 2 \frac{d}{d x} [x] + \frac{d}{d x} [- 3])$

Step 1.1.3.5

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

$4 {(x^{2} - 2 x - 3)}^{- 2} (2 x - 2 \cdot 1 + \frac{d}{d x} [- 3])$

Step 1.1.3.6

Multiply $- 2$ by $1$ .

$4 {(x^{2} - 2 x - 3)}^{- 2} (2 x - 2 + \frac{d}{d x} [- 3])$

Step 1.1.3.7

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

$4 {(x^{2} - 2 x - 3)}^{- 2} (2 x - 2 + 0)$

Step 1.1.3.8

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

$4 {(x^{2} - 2 x - 3)}^{- 2} (2 x - 2)$

Step 1.1.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$4 \frac{1}{{(x^{2} - 2 x - 3)}^{2}} (2 x - 2)$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Combine $4$ and $\frac{1}{{(x^{2} - 2 x - 3)}^{2}}$ .

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

Step 1.1.5.2

Reorder the factors of $\frac{4}{{(x^{2} - 2 x - 3)}^{2}} (2 x - 2)$ .

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

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $(2 x - 2) (\frac{4}{{(x^{2} - 2 x - 3)}^{2}})$ .

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

Step 2

Set the first derivative equal to $0$ then solve the equation $(2 x - 2) (\frac{4}{{(x^{2} - 2 x - 3)}^{2}}) = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

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

$2 x - 2 = 0$

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

Step 2.3

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

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Step 2.3.1

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

$2 x - 2 = 0$

Step 2.3.2

Solve $2 x - 2 = 0$ for $x$ .

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Step 2.3.2.1

Add $2$ to both sides of the equation.

$2 x = 2$

Step 2.3.2.2

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

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Step 2.3.2.2.1

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

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

Step 2.3.2.2.2

Simplify the left side.

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Step 2.3.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.2.2.1.1

Cancel the common factor.

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

Step 2.3.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.3.2.2.3

Simplify the right side.

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Step 2.3.2.2.3.1

Divide $2$ by $2$ .

$x = 1$

Step 2.4

Set $\frac{4}{{(x^{2} - 2 x - 3)}^{2}}$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $\frac{4}{{(x^{2} - 2 x - 3)}^{2}}$ equal to $0$ .

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

Step 2.4.2

Solve $\frac{4}{{(x^{2} - 2 x - 3)}^{2}} = 0$ for $x$ .

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Step 2.4.2.1

Set the numerator equal to zero.

$4 = 0$

Step 2.4.2.2

Since $4 \neq 0$ , there are no solutions.

No solution

Step 2.5

The final solution is all the values that make $(2 x - 2) (\frac{4}{{(x^{2} - 2 x - 3)}^{2}}) = 0$ true.

$x = 1$

Step 3

The values which make the derivative equal to $0$ are $1$ .

$1$

Step 4

Find where the derivative is undefined.

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Step 4.1

Set the denominator in $\frac{4}{{(x^{2} - 2 x - 3)}^{2}}$ equal to $0$ to find where the expression is undefined.

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

Step 4.2

Solve for $x$ .

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Step 4.2.1

Factor the left side of the equation.

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Step 4.2.1.1

Factor $x^{2} - 2 x - 3$ using the AC method.

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Step 4.2.1.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 4.2.1.1.2

Write the factored form using these integers.

${((x - 3) (x + 1))}^{2} = 0$

Step 4.2.1.2

Apply the product rule to $(x - 3) (x + 1)$ .

${(x - 3)}^{2} {(x + 1)}^{2} = 0$

Step 4.2.2

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

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

${(x + 1)}^{2} = 0$

Step 4.2.3

Set ${(x - 3)}^{2}$ equal to $0$ and solve for $x$ .

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Step 4.2.3.1

Set ${(x - 3)}^{2}$ equal to $0$ .

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

Step 4.2.3.2

Solve ${(x - 3)}^{2} = 0$ for $x$ .

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Step 4.2.3.2.1

Set the $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 4.2.3.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 4.2.4

Set ${(x + 1)}^{2}$ equal to $0$ and solve for $x$ .

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Step 4.2.4.1

Set ${(x + 1)}^{2}$ equal to $0$ .

${(x + 1)}^{2} = 0$

Step 4.2.4.2

Solve ${(x + 1)}^{2} = 0$ for $x$ .

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Step 4.2.4.2.1

Set the $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 4.2.4.2.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 4.2.5

The final solution is all the values that make ${(x - 3)}^{2} {(x + 1)}^{2} = 0$ true.

$x = 3, - 1$

Step 4.3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 1, x = 3$

Step 5

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, - 1) \cup (- 1, 1) \cup (1, 3) \cup (3, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 1)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 2$ in the expression.

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

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the denominator.

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Step 6.2.1.1

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

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

Step 6.2.1.2

Multiply $- 2$ by $- 2$ .

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

Step 6.2.1.3

Add $4$ and $4$ .

$f' (- 2) = (2 (- 2) - 2) (\frac{4}{{(8 - 3)}^{2}})$

Step 6.2.1.4

Subtract $3$ from $8$ .

$f' (- 2) = (2 (- 2) - 2) (\frac{4}{5^{2}})$

Step 6.2.1.5

Raise $5$ to the power of $2$ .

$f' (- 2) = (2 (- 2) - 2) (\frac{4}{25})$

Step 6.2.2

Simplify the expression.

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Step 6.2.2.1

Multiply $2$ by $- 2$ .

$f' (- 2) = (- 4 - 2) (\frac{4}{25})$

Step 6.2.2.2

Subtract $2$ from $- 4$ .

$f' (- 2) = - 6 (\frac{4}{25})$

Step 6.2.3

Multiply $- 6 (\frac{4}{25})$ .

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Step 6.2.3.1

Combine $- 6$ and $\frac{4}{25}$ .

$f' (- 2) = \frac{- 6 \cdot 4}{25}$

Step 6.2.3.2

Multiply $- 6$ by $4$ .

$f' (- 2) = \frac{- 24}{25}$

Step 6.2.4

Move the negative in front of the fraction.

$f' (- 2) = - \frac{24}{25}$

Step 6.2.5

The final answer is $- \frac{24}{25}$ .

$- \frac{24}{25}$

Step 6.3

At $x = - 2$ the derivative is $- \frac{24}{25}$ . Since this is negative, the function is decreasing on $(- \infty, - 1)$ .

Decreasing on $(- \infty, - 1)$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(- 1, 1)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $0$ in the expression.

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

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify the denominator.

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Step 7.2.1.1

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

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

Step 7.2.1.2

Multiply $- 2$ by $0$ .

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

Step 7.2.1.3

Add $0$ and $0$ .

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

Step 7.2.1.4

Subtract $3$ from $0$ .

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

Step 7.2.1.5

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

$f' (0) = (2 (0) - 2) (\frac{4}{9})$

Step 7.2.2

Simplify the expression.

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Step 7.2.2.1

Multiply $2$ by $0$ .

$f' (0) = (0 - 2) (\frac{4}{9})$

Step 7.2.2.2

Subtract $2$ from $0$ .

$f' (0) = - 2 (\frac{4}{9})$

Step 7.2.3

Multiply $- 2 (\frac{4}{9})$ .

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Step 7.2.3.1

Combine $- 2$ and $\frac{4}{9}$ .

$f' (0) = \frac{- 2 \cdot 4}{9}$

Step 7.2.3.2

Multiply $- 2$ by $4$ .

$f' (0) = \frac{- 8}{9}$

Step 7.2.4

Move the negative in front of the fraction.

$f' (0) = - \frac{8}{9}$

Step 7.2.5

The final answer is $- \frac{8}{9}$ .

$- \frac{8}{9}$

Step 7.3

At $x = 0$ the derivative is $- \frac{8}{9}$ . Since this is negative, the function is decreasing on $(- 1, 1)$ .

Decreasing on $(- 1, 1)$ since $f' (x) < 0$

Step 8

Substitute a value from the interval $(1, 3)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $2$ in the expression.

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

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify the denominator.

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Step 8.2.1.1

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

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

Step 8.2.1.2

Multiply $- 2$ by $2$ .

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

Step 8.2.1.3

Subtract $4$ from $4$ .

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

Step 8.2.1.4

Subtract $3$ from $0$ .

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

Step 8.2.1.5

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

$f' (2) = (2 (2) - 2) (\frac{4}{9})$

Step 8.2.2

Simplify the expression.

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Step 8.2.2.1

Multiply $2$ by $2$ .

$f' (2) = (4 - 2) (\frac{4}{9})$

Step 8.2.2.2

Subtract $2$ from $4$ .

$f' (2) = 2 (\frac{4}{9})$

Step 8.2.3

Multiply $2 (\frac{4}{9})$ .

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Step 8.2.3.1

Combine $2$ and $\frac{4}{9}$ .

$f' (2) = \frac{2 \cdot 4}{9}$

Step 8.2.3.2

Multiply $2$ by $4$ .

$f' (2) = \frac{8}{9}$

Step 8.2.4

The final answer is $\frac{8}{9}$ .

$\frac{8}{9}$

Step 8.3

At $x = 2$ the derivative is $\frac{8}{9}$ . Since this is positive, the function is increasing on $(1, 3)$ .

Increasing on $(1, 3)$ since $f' (x) > 0$

Step 9

Substitute a value from the interval $(3, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 9.1

Replace the variable $x$ with $4$ in the expression.

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

Step 9.2

Simplify the result.

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Step 9.2.1

Simplify the denominator.

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Step 9.2.1.1

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

$f' (4) = (2 (4) - 2) (\frac{4}{{(16 - 2 \cdot 4 - 3)}^{2}})$

Step 9.2.1.2

Multiply $- 2$ by $4$ .

$f' (4) = (2 (4) - 2) (\frac{4}{{(16 - 8 - 3)}^{2}})$

Step 9.2.1.3

Subtract $8$ from $16$ .

$f' (4) = (2 (4) - 2) (\frac{4}{{(8 - 3)}^{2}})$

Step 9.2.1.4

Subtract $3$ from $8$ .

$f' (4) = (2 (4) - 2) (\frac{4}{5^{2}})$

Step 9.2.1.5

Raise $5$ to the power of $2$ .

$f' (4) = (2 (4) - 2) (\frac{4}{25})$

Step 9.2.2

Simplify the expression.

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Step 9.2.2.1

Multiply $2$ by $4$ .

$f' (4) = (8 - 2) (\frac{4}{25})$

Step 9.2.2.2

Subtract $2$ from $8$ .

$f' (4) = 6 (\frac{4}{25})$

Step 9.2.3

Multiply $6 (\frac{4}{25})$ .

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Step 9.2.3.1

Combine $6$ and $\frac{4}{25}$ .

$f' (4) = \frac{6 \cdot 4}{25}$

Step 9.2.3.2

Multiply $6$ by $4$ .

$f' (4) = \frac{24}{25}$

Step 9.2.4

The final answer is $\frac{24}{25}$ .

$\frac{24}{25}$

Step 9.3

At $x = 4$ the derivative is $\frac{24}{25}$ . Since this is positive, the function is increasing on $(3, \infty)$ .

Increasing on $(3, \infty)$ since $f' (x) > 0$

Step 10

List the intervals on which the function is increasing and decreasing.

Increasing on: $(1, 3), (3, \infty)$

Decreasing on: $(- \infty, - 1), (- 1, 1)$

Step 11