Calculus Examples

$f (x) = \frac{x + 2}{x^{2} - 3 x - 10}$

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 $f (x) = x + 2$ and $g (x) = x^{2} - 3 x - 10$ .

$\frac{(x^{2} - 3 x - 10) \frac{d}{d x} [x + 2] - (x + 2) \frac{d}{d x} [x^{2} - 3 x - 10]}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

$\frac{(x^{2} - 3 x - 10) (\frac{d}{d x} [x] + \frac{d}{d x} [2]) - (x + 2) \frac{d}{d x} [x^{2} - 3 x - 10]}{{(x^{2} - 3 x - 10)}^{2}}$

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 $n = 1$ .

$\frac{(x^{2} - 3 x - 10) (1 + \frac{d}{d x} [2]) - (x + 2) \frac{d}{d x} [x^{2} - 3 x - 10]}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.2.3

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

$\frac{(x^{2} - 3 x - 10) (1 + 0) - (x + 2) \frac{d}{d x} [x^{2} - 3 x - 10]}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.2.4

Simplify the expression.

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Step 1.1.2.4.1

Add $1$ and $0$ .

$\frac{(x^{2} - 3 x - 10) \cdot 1 - (x + 2) \frac{d}{d x} [x^{2} - 3 x - 10]}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.2.4.2

Multiply $x^{2} - 3 x - 10$ by $1$ .

$\frac{x^{2} - 3 x - 10 - (x + 2) \frac{d}{d x} [x^{2} - 3 x - 10]}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.2.5

By the Sum Rule, the derivative of $x^{2} - 3 x - 10$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 10]$ .

$\frac{x^{2} - 3 x - 10 - (x + 2) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 10])}{{(x^{2} - 3 x - 10)}^{2}}$

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 $n = 2$ .

$\frac{x^{2} - 3 x - 10 - (x + 2) (2 x + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 10])}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.2.7

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

$\frac{x^{2} - 3 x - 10 - (x + 2) (2 x - 3 \frac{d}{d x} [x] + \frac{d}{d x} [- 10])}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.2.8

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

$\frac{x^{2} - 3 x - 10 - (x + 2) (2 x - 3 \cdot 1 + \frac{d}{d x} [- 10])}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.2.9

Multiply $- 3$ by $1$ .

$\frac{x^{2} - 3 x - 10 - (x + 2) (2 x - 3 + \frac{d}{d x} [- 10])}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.2.10

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10$ with respect to $x$ is $0$ .

$\frac{x^{2} - 3 x - 10 - (x + 2) (2 x - 3 + 0)}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.2.11

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

$\frac{x^{2} - 3 x - 10 - (x + 2) (2 x - 3)}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.3

Simplify.

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Step 1.1.3.1

Apply the distributive property.

$\frac{x^{2} - 3 x - 10 + (- x - 1 \cdot 2) (2 x - 3)}{{(x^{2} - 3 x - 10)}^{2}}$

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

Multiply $- 1$ by $2$ .

$\frac{x^{2} - 3 x - 10 + (- x - 2) (2 x - 3)}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.3.2.1.2

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

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Step 1.1.3.2.1.2.1

Apply the distributive property.

$\frac{x^{2} - 3 x - 10 - x (2 x - 3) - 2 (2 x - 3)}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.3.2.1.2.2

Apply the distributive property.

$\frac{x^{2} - 3 x - 10 - x (2 x) - x \cdot - 3 - 2 (2 x - 3)}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.3.2.1.2.3

Apply the distributive property.

$\frac{x^{2} - 3 x - 10 - x (2 x) - x \cdot - 3 - 2 (2 x) - 2 \cdot - 3}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.3.2.1.3

Simplify and combine like terms.

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Step 1.1.3.2.1.3.1

Simplify each term.

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Step 1.1.3.2.1.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{x^{2} - 3 x - 10 - 1 \cdot 2 x \cdot x - x \cdot - 3 - 2 (2 x) - 2 \cdot - 3}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.3.2.1.3.1.2

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

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Step 1.1.3.2.1.3.1.2.1

Move $x$ .

$\frac{x^{2} - 3 x - 10 - 1 \cdot 2 (x \cdot x) - x \cdot - 3 - 2 (2 x) - 2 \cdot - 3}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.3.2.1.3.1.2.2

Multiply $x$ by $x$ .

$\frac{x^{2} - 3 x - 10 - 1 \cdot 2 x^{2} - x \cdot - 3 - 2 (2 x) - 2 \cdot - 3}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.3.2.1.3.1.3

Multiply $- 1$ by $2$ .

$\frac{x^{2} - 3 x - 10 - 2 x^{2} - x \cdot - 3 - 2 (2 x) - 2 \cdot - 3}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.3.2.1.3.1.4

Multiply $- 3$ by $- 1$ .

$\frac{x^{2} - 3 x - 10 - 2 x^{2} + 3 x - 2 (2 x) - 2 \cdot - 3}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.3.2.1.3.1.5

Multiply $2$ by $- 2$ .

$\frac{x^{2} - 3 x - 10 - 2 x^{2} + 3 x - 4 x - 2 \cdot - 3}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.3.2.1.3.1.6

Multiply $- 2$ by $- 3$ .

$\frac{x^{2} - 3 x - 10 - 2 x^{2} + 3 x - 4 x + 6}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.3.2.1.3.2

Subtract $4 x$ from $3 x$ .

$\frac{x^{2} - 3 x - 10 - 2 x^{2} - x + 6}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.3.2.2

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

$\frac{- x^{2} - 3 x - 10 - x + 6}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.3.2.3

Subtract $x$ from $- 3 x$ .

$\frac{- x^{2} - 4 x - 10 + 6}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.3.2.4

Add $- 10$ and $6$ .

$\frac{- x^{2} - 4 x - 4}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.3.3

Factor by grouping.

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Step 1.1.3.3.1

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 $a \cdot c = - 1 \cdot - 4 = 4$ and whose sum is $b = - 4$ .

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Step 1.1.3.3.1.1

Factor $- 4$ out of $- 4 x$ .

$\frac{- x^{2} - 4 (x) - 4}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.3.3.1.2

Rewrite $- 4$ as $- 2$ plus $- 2$

$\frac{- x^{2} + (- 2 - 2) x - 4}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.3.3.1.3

Apply the distributive property.

$\frac{- x^{2} - 2 x - 2 x - 4}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.3.3.2

Factor out the greatest common factor from each group.

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Step 1.1.3.3.2.1

Group the first two terms and the last two terms.

$\frac{(- x^{2} - 2 x) - 2 x - 4}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.3.3.2.2

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

$\frac{x (- x - 2) + 2 (- x - 2)}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.3.3.3

Factor the polynomial by factoring out the greatest common factor, $- x - 2$ .

$\frac{(- x - 2) (x + 2)}{{(x^{2} - 3 x - 10)}^{2}}$

Step 1.1.3.4

Simplify the denominator.

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Step 1.1.3.4.1

Factor $x^{2} - 3 x - 10$ using the AC method.

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Step 1.1.3.4.1.1

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 $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Step 1.1.3.4.1.2

Write the factored form using these integers.

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

Step 1.1.3.4.2

Apply the product rule to $(x - 5) (x + 2)$ .

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

Step 1.1.3.5

Simplify the numerator.

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Step 1.1.3.5.1

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

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

Step 1.1.3.5.2

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 1.1.3.5.3

Factor $- 1$ out of $- (x) - 1 (2)$ .

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

Step 1.1.3.5.4

Rewrite $- (x + 2)$ as $- 1 (x + 2)$ .

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

Step 1.1.3.5.5

Raise $x + 2$ to the power of $1$ .

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

Step 1.1.3.5.6

Raise $x + 2$ to the power of $1$ .

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

Step 1.1.3.5.7

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

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

Step 1.1.3.5.8

Add $1$ and $1$ .

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

Step 1.1.3.6

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

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Step 1.1.3.6.1

Cancel the common factor.

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

Step 1.1.3.6.2

Rewrite the expression.

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

Step 1.1.3.7

Move the negative in front of the fraction.

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

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{1}{{(x - 5)}^{2}}$ .

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

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{1}{{(x - 5)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

$1 = 0$

Step 2.3

Since $1 \neq 0$ , there are no solutions.

No solution

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{1}{{(x - 5)}^{2}}$ equal to $0$ to find where the expression is undefined.

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

Step 3.2

Solve for $x$ .

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Step 3.2.1

Set the $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 3.2.2

Add $5$ to both sides of the equation.

$x = 5$

Step 4

Evaluate $\frac{x + 2}{x^{2} - 3 x - 10}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 5$ .

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Step 4.1.1

Substitute $5$ for $x$ .

$\frac{(5) + 2}{{(5)}^{2} - 3 \cdot 5 - 10}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

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

$\frac{(5) + 2}{25 - 3 \cdot 5 - 10}$

Step 4.1.2.1.2

Multiply $- 3$ by $5$ .

$\frac{(5) + 2}{25 - 15 - 10}$

Step 4.1.2.2

Simplify by subtracting numbers.

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Step 4.1.2.2.1

Subtract $15$ from $25$ .

$\frac{(5) + 2}{10 - 10}$

Step 4.1.2.2.2

Subtract $10$ from $10$ .

$\frac{(5) + 2}{0}$

Step 4.1.2.2.3

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

Undefined

$\frac{(5) + 2}{0}$

Step 4.1.2.3

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