Calculus Examples

$f (x) = \frac{x + 3}{x^{2} - 2 x - 15}$

Step 1

Find the first derivative of the function.

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Step 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 $g (x) = x^{2} - 2 x - 15$ .

$\frac{(x^{2} - 2 x - 15) \frac{d}{d x} [x + 3] - (x + 3) \frac{d}{d x} [x^{2} - 2 x - 15]}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.2

Differentiate.

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Step 1.2.1

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

$\frac{(x^{2} - 2 x - 15) (\frac{d}{d x} [x] + \frac{d}{d x} [3]) - (x + 3) \frac{d}{d x} [x^{2} - 2 x - 15]}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.2.2

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

$\frac{(x^{2} - 2 x - 15) (1 + \frac{d}{d x} [3]) - (x + 3) \frac{d}{d x} [x^{2} - 2 x - 15]}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.2.3

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

$\frac{(x^{2} - 2 x - 15) (1 + 0) - (x + 3) \frac{d}{d x} [x^{2} - 2 x - 15]}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Add $1$ and $0$ .

$\frac{(x^{2} - 2 x - 15) \cdot 1 - (x + 3) \frac{d}{d x} [x^{2} - 2 x - 15]}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.2.4.2

Multiply $x^{2} - 2 x - 15$ by $1$ .

$\frac{x^{2} - 2 x - 15 - (x + 3) \frac{d}{d x} [x^{2} - 2 x - 15]}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.2.5

By the Sum Rule, the derivative of $x^{2} - 2 x - 15$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 15]$ .

$\frac{x^{2} - 2 x - 15 - (x + 3) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 15])}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.2.6

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

$\frac{x^{2} - 2 x - 15 - (x + 3) (2 x + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 15])}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.2.7

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

$\frac{x^{2} - 2 x - 15 - (x + 3) (2 x - 2 \frac{d}{d x} [x] + \frac{d}{d x} [- 15])}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.2.8

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

$\frac{x^{2} - 2 x - 15 - (x + 3) (2 x - 2 \cdot 1 + \frac{d}{d x} [- 15])}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.2.9

Multiply $- 2$ by $1$ .

$\frac{x^{2} - 2 x - 15 - (x + 3) (2 x - 2 + \frac{d}{d x} [- 15])}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.2.10

Since $- 15$ is constant with respect to $x$ , the derivative of $- 15$ with respect to $x$ is $0$ .

$\frac{x^{2} - 2 x - 15 - (x + 3) (2 x - 2 + 0)}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.2.11

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

$\frac{x^{2} - 2 x - 15 - (x + 3) (2 x - 2)}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

$\frac{x^{2} - 2 x - 15 + (- x - 1 \cdot 3) (2 x - 2)}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.3.2

Simplify the numerator.

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Step 1.3.2.1

Simplify each term.

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Step 1.3.2.1.1

Multiply $- 1$ by $3$ .

$\frac{x^{2} - 2 x - 15 + (- x - 3) (2 x - 2)}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.3.2.1.2

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

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Step 1.3.2.1.2.1

Apply the distributive property.

$\frac{x^{2} - 2 x - 15 - x (2 x - 2) - 3 (2 x - 2)}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.3.2.1.2.2

Apply the distributive property.

$\frac{x^{2} - 2 x - 15 - x (2 x) - x \cdot - 2 - 3 (2 x - 2)}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.3.2.1.2.3

Apply the distributive property.

$\frac{x^{2} - 2 x - 15 - x (2 x) - x \cdot - 2 - 3 (2 x) - 3 \cdot - 2}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.3.2.1.3

Simplify and combine like terms.

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Step 1.3.2.1.3.1

Simplify each term.

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Step 1.3.2.1.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{x^{2} - 2 x - 15 - 1 \cdot 2 x \cdot x - x \cdot - 2 - 3 (2 x) - 3 \cdot - 2}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.3.2.1.3.1.2

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

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Step 1.3.2.1.3.1.2.1

Move $x$ .

$\frac{x^{2} - 2 x - 15 - 1 \cdot 2 (x \cdot x) - x \cdot - 2 - 3 (2 x) - 3 \cdot - 2}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.3.2.1.3.1.2.2

Multiply $x$ by $x$ .

$\frac{x^{2} - 2 x - 15 - 1 \cdot 2 x^{2} - x \cdot - 2 - 3 (2 x) - 3 \cdot - 2}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.3.2.1.3.1.3

Multiply $- 1$ by $2$ .

$\frac{x^{2} - 2 x - 15 - 2 x^{2} - x \cdot - 2 - 3 (2 x) - 3 \cdot - 2}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.3.2.1.3.1.4

Multiply $- 2$ by $- 1$ .

$\frac{x^{2} - 2 x - 15 - 2 x^{2} + 2 x - 3 (2 x) - 3 \cdot - 2}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.3.2.1.3.1.5

Multiply $2$ by $- 3$ .

$\frac{x^{2} - 2 x - 15 - 2 x^{2} + 2 x - 6 x - 3 \cdot - 2}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.3.2.1.3.1.6

Multiply $- 3$ by $- 2$ .

$\frac{x^{2} - 2 x - 15 - 2 x^{2} + 2 x - 6 x + 6}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.3.2.1.3.2

Subtract $6 x$ from $2 x$ .

$\frac{x^{2} - 2 x - 15 - 2 x^{2} - 4 x + 6}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.3.2.2

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

$\frac{- x^{2} - 2 x - 15 - 4 x + 6}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.3.2.3

Subtract $4 x$ from $- 2 x$ .

$\frac{- x^{2} - 6 x - 15 + 6}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.3.2.4

Add $- 15$ and $6$ .

$\frac{- x^{2} - 6 x - 9}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.3.3

Factor by grouping.

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Step 1.3.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 9 = 9$ and whose sum is $b = - 6$ .

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Step 1.3.3.1.1

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

$\frac{- x^{2} - 6 (x) - 9}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.3.3.1.2

Rewrite $- 6$ as $- 3$ plus $- 3$

$\frac{- x^{2} + (- 3 - 3) x - 9}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.3.3.1.3

Apply the distributive property.

$\frac{- x^{2} - 3 x - 3 x - 9}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.3.3.2

Factor out the greatest common factor from each group.

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Step 1.3.3.2.1

Group the first two terms and the last two terms.

$\frac{(- x^{2} - 3 x) - 3 x - 9}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.3.3.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{x (- x - 3) + 3 (- x - 3)}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.3.3.3

Factor the polynomial by factoring out the greatest common factor, $- x - 3$ .

$\frac{(- x - 3) (x + 3)}{{(x^{2} - 2 x - 15)}^{2}}$

Step 1.3.4

Simplify the denominator.

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Step 1.3.4.1

Factor $x^{2} - 2 x - 15$ using the AC method.

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Step 1.3.4.1.1

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 $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Step 1.3.4.1.2

Write the factored form using these integers.

$\frac{(- x - 3) (x + 3)}{{((x - 5) (x + 3))}^{2}}$

Step 1.3.4.2

Apply the product rule to $(x - 5) (x + 3)$ .

$\frac{(- x - 3) (x + 3)}{{(x - 5)}^{2} {(x + 3)}^{2}}$

Step 1.3.5

Simplify the numerator.

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Step 1.3.5.1

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

$\frac{(- (x) - 3) (x + 3)}{{(x - 5)}^{2} {(x + 3)}^{2}}$

Step 1.3.5.2

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{(- (x) - 1 (3)) (x + 3)}{{(x - 5)}^{2} {(x + 3)}^{2}}$

Step 1.3.5.3

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

$\frac{- (x + 3) (x + 3)}{{(x - 5)}^{2} {(x + 3)}^{2}}$

Step 1.3.5.4

Rewrite $- (x + 3)$ as $- 1 (x + 3)$ .

$\frac{- 1 (x + 3) (x + 3)}{{(x - 5)}^{2} {(x + 3)}^{2}}$

Step 1.3.5.5

Raise $x + 3$ to the power of $1$ .

$\frac{- 1 ({(x + 3)}^{1} (x + 3))}{{(x - 5)}^{2} {(x + 3)}^{2}}$

Step 1.3.5.6

Raise $x + 3$ to the power of $1$ .

$\frac{- 1 ({(x + 3)}^{1} {(x + 3)}^{1})}{{(x - 5)}^{2} {(x + 3)}^{2}}$

Step 1.3.5.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{- 1 {(x + 3)}^{1 + 1}}{{(x - 5)}^{2} {(x + 3)}^{2}}$

Step 1.3.5.8

Add $1$ and $1$ .

$\frac{- 1 {(x + 3)}^{2}}{{(x - 5)}^{2} {(x + 3)}^{2}}$

Step 1.3.6

Cancel the common factor of ${(x + 3)}^{2}$ .

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Step 1.3.6.1

Cancel the common factor.

$\frac{- 1 {(x + 3)}^{2}}{{(x - 5)}^{2} {(x + 3)}^{2}}$

Step 1.3.6.2

Rewrite the expression.

$\frac{- 1}{{(x - 5)}^{2}}$

Step 1.3.7

Move the negative in front of the fraction.

$- \frac{1}{{(x - 5)}^{2}}$

Step 2

Find the second derivative of the function.

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Step 2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{1}{{(x - 5)}^{2}}$ .

$f'' (x) = - \frac{d}{d x} \cdot \frac{1}{{(x - 5)}^{2}} + \frac{1}{{(x - 5)}^{2}} \cdot \frac{d}{d x} (- 1)$

Step 2.2

Apply basic rules of exponents.

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Step 2.2.1

Rewrite $\frac{1}{{(x - 5)}^{2}}$ as ${({(x - 5)}^{2})}^{- 1}$ .

$f'' (x) = - \frac{d}{d x} {({(x - 5)}^{2})}^{- 1} + \frac{1}{{(x - 5)}^{2}} \cdot \frac{d}{d x} (- 1)$

Step 2.2.2

Multiply the exponents in ${({(x - 5)}^{2})}^{- 1}$ .

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Step 2.2.2.1

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

$f'' (x) = - \frac{d}{d x} {(x - 5)}^{2 \cdot - 1} + \frac{1}{{(x - 5)}^{2}} \cdot \frac{d}{d x} (- 1)$

Step 2.2.2.2

Multiply $2$ by $- 1$ .

$f'' (x) = - \frac{d}{d x} {(x - 5)}^{- 2} + \frac{1}{{(x - 5)}^{2}} \cdot \frac{d}{d x} (- 1)$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 2}$ and $g (x) = x - 5$ .

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Step 2.3.1

To apply the Chain Rule, set $u$ as $x - 5$ .

$f'' (x) = - (\frac{d}{d u} (u^{- 2}) \frac{d}{d x} (x - 5)) + \frac{1}{{(x - 5)}^{2}} \cdot \frac{d}{d x} (- 1)$

Step 2.3.2

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

$f'' (x) = - (- 2 u^{- 3} \frac{d}{d x} (x - 5)) + \frac{1}{{(x - 5)}^{2}} \cdot \frac{d}{d x} (- 1)$

Step 2.3.3

Replace all occurrences of $u$ with $x - 5$ .

$f'' (x) = - (- 2 {(x - 5)}^{- 3} \frac{d}{d x} (x - 5)) + \frac{1}{{(x - 5)}^{2}} \cdot \frac{d}{d x} (- 1)$

Step 2.4

Differentiate.

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Step 2.4.1

Multiply $- 2$ by $- 1$ .

$f'' (x) = 2 ({(x - 5)}^{- 3} \frac{d}{d x} (x - 5)) + \frac{1}{{(x - 5)}^{2}} \cdot \frac{d}{d x} (- 1)$

Step 2.4.2

By the Sum Rule, the derivative of $x - 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 5]$ .

$f'' (x) = 2 {(x - 5)}^{- 3} (\frac{d}{d x} (x) + \frac{d}{d x} (- 5)) + \frac{1}{{(x - 5)}^{2}} \cdot \frac{d}{d x} (- 1)$

Step 2.4.3

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

$f'' (x) = 2 {(x - 5)}^{- 3} (1 + \frac{d}{d x} (- 5)) + \frac{1}{{(x - 5)}^{2}} \cdot \frac{d}{d x} (- 1)$

Step 2.4.4

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$f'' (x) = 2 {(x - 5)}^{- 3} (1 + 0) + \frac{1}{{(x - 5)}^{2}} \cdot \frac{d}{d x} (- 1)$

Step 2.4.5

Simplify the expression.

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Step 2.4.5.1

Add $1$ and $0$ .

$f'' (x) = 2 {(x - 5)}^{- 3} \cdot 1 + \frac{1}{{(x - 5)}^{2}} \cdot \frac{d}{d x} (- 1)$

Step 2.4.5.2

Multiply $2$ by $1$ .

$f'' (x) = 2 {(x - 5)}^{- 3} + \frac{1}{{(x - 5)}^{2}} \cdot \frac{d}{d x} (- 1)$

Step 2.4.6

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$f'' (x) = 2 {(x - 5)}^{- 3} + \frac{1}{{(x - 5)}^{2}} \cdot 0$

Step 2.4.7

Simplify the expression.

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Step 2.4.7.1

Multiply $\frac{1}{{(x - 5)}^{2}}$ by $0$ .

$f'' (x) = 2 {(x - 5)}^{- 3} + 0$

Step 2.4.7.2

Add $2 {(x - 5)}^{- 3}$ and $0$ .

$f'' (x) = 2 {(x - 5)}^{- 3}$

Step 2.5

Simplify.

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Step 2.5.1

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

$f'' (x) = 2 (\frac{1}{{(x - 5)}^{3}})$

Step 2.5.2

Combine $2$ and $\frac{1}{{(x - 5)}^{3}}$ .

$f'' (x) = \frac{2}{{(x - 5)}^{3}}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$- \frac{1}{{(x - 5)}^{2}} = 0$

Step 4

Since there is no value of $x$ that makes the first derivative equal to $0$ , there are no local extrema.

No Local Extrema

Step 5

No Local Extrema

Step 6