Calculus Examples

$f (x) = \frac{2 x + 5}{- 6 x - 8}$

Step 1

Find the first derivative.

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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 $f (x) = 2 x + 5$ and $g (x) = - 6 x - 8$ .

$\frac{(- 6 x - 8) \frac{d}{d x} [2 x + 5] - (2 x + 5) \frac{d}{d x} [- 6 x - 8]}{{(- 6 x - 8)}^{2}}$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $2 x + 5$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [5]$ .

$\frac{(- 6 x - 8) (\frac{d}{d x} [2 x] + \frac{d}{d x} [5]) - (2 x + 5) \frac{d}{d x} [- 6 x - 8]}{{(- 6 x - 8)}^{2}}$

Step 1.2.2

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

$\frac{(- 6 x - 8) (2 \frac{d}{d x} [x] + \frac{d}{d x} [5]) - (2 x + 5) \frac{d}{d x} [- 6 x - 8]}{{(- 6 x - 8)}^{2}}$

Step 1.2.3

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

$\frac{(- 6 x - 8) (2 \cdot 1 + \frac{d}{d x} [5]) - (2 x + 5) \frac{d}{d x} [- 6 x - 8]}{{(- 6 x - 8)}^{2}}$

Step 1.2.4

Multiply $2$ by $1$ .

$\frac{(- 6 x - 8) (2 + \frac{d}{d x} [5]) - (2 x + 5) \frac{d}{d x} [- 6 x - 8]}{{(- 6 x - 8)}^{2}}$

Step 1.2.5

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

$\frac{(- 6 x - 8) (2 + 0) - (2 x + 5) \frac{d}{d x} [- 6 x - 8]}{{(- 6 x - 8)}^{2}}$

Step 1.2.6

Simplify the expression.

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Step 1.2.6.1

Add $2$ and $0$ .

$\frac{(- 6 x - 8) \cdot 2 - (2 x + 5) \frac{d}{d x} [- 6 x - 8]}{{(- 6 x - 8)}^{2}}$

Step 1.2.6.2

Move $2$ to the left of $- 6 x - 8$ .

$\frac{2 \cdot (- 6 x - 8) - (2 x + 5) \frac{d}{d x} [- 6 x - 8]}{{(- 6 x - 8)}^{2}}$

Step 1.2.7

By the Sum Rule, the derivative of $- 6 x - 8$ with respect to $x$ is $\frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 8]$ .

$\frac{2 (- 6 x - 8) - (2 x + 5) (\frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 8])}{{(- 6 x - 8)}^{2}}$

Step 1.2.8

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

$\frac{2 (- 6 x - 8) - (2 x + 5) (- 6 \frac{d}{d x} [x] + \frac{d}{d x} [- 8])}{{(- 6 x - 8)}^{2}}$

Step 1.2.9

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

$\frac{2 (- 6 x - 8) - (2 x + 5) (- 6 \cdot 1 + \frac{d}{d x} [- 8])}{{(- 6 x - 8)}^{2}}$

Step 1.2.10

Multiply $- 6$ by $1$ .

$\frac{2 (- 6 x - 8) - (2 x + 5) (- 6 + \frac{d}{d x} [- 8])}{{(- 6 x - 8)}^{2}}$

Step 1.2.11

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$\frac{2 (- 6 x - 8) - (2 x + 5) (- 6 + 0)}{{(- 6 x - 8)}^{2}}$

Step 1.2.12

Simplify the expression.

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Step 1.2.12.1

Add $- 6$ and $0$ .

$\frac{2 (- 6 x - 8) - (2 x + 5) \cdot - 6}{{(- 6 x - 8)}^{2}}$

Step 1.2.12.2

Multiply $- 6$ by $- 1$ .

$\frac{2 (- 6 x - 8) + 6 (2 x + 5)}{{(- 6 x - 8)}^{2}}$

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

$\frac{2 (- 6 x) + 2 \cdot - 8 + 6 (2 x + 5)}{{(- 6 x - 8)}^{2}}$

Step 1.3.2

Apply the distributive property.

$\frac{2 (- 6 x) + 2 \cdot - 8 + 6 (2 x) + 6 \cdot 5}{{(- 6 x - 8)}^{2}}$

Step 1.3.3

Simplify the numerator.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Multiply $- 6$ by $2$ .

$\frac{- 12 x + 2 \cdot - 8 + 6 (2 x) + 6 \cdot 5}{{(- 6 x - 8)}^{2}}$

Step 1.3.3.1.2

Multiply $2$ by $- 8$ .

$\frac{- 12 x - 16 + 6 (2 x) + 6 \cdot 5}{{(- 6 x - 8)}^{2}}$

Step 1.3.3.1.3

Multiply $2$ by $6$ .

$\frac{- 12 x - 16 + 12 x + 6 \cdot 5}{{(- 6 x - 8)}^{2}}$

Step 1.3.3.1.4

Multiply $6$ by $5$ .

$\frac{- 12 x - 16 + 12 x + 30}{{(- 6 x - 8)}^{2}}$

Step 1.3.3.2

Combine the opposite terms in $- 12 x - 16 + 12 x + 30$ .

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Step 1.3.3.2.1

Add $- 12 x$ and $12 x$ .

$\frac{0 - 16 + 30}{{(- 6 x - 8)}^{2}}$

Step 1.3.3.2.2

Subtract $16$ from $0$ .

$\frac{- 16 + 30}{{(- 6 x - 8)}^{2}}$

Step 1.3.3.3

Add $- 16$ and $30$ .

$\frac{14}{{(- 6 x - 8)}^{2}}$

Step 1.3.4

Simplify the denominator.

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Step 1.3.4.1

Factor $2$ out of $- 6 x - 8$ .

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Step 1.3.4.1.1

Factor $2$ out of $- 6 x$ .

$\frac{14}{{(2 (- 3 x) - 8)}^{2}}$

Step 1.3.4.1.2

Factor $2$ out of $- 8$ .

$\frac{14}{{(2 (- 3 x) + 2 (- 4))}^{2}}$

Step 1.3.4.1.3

Factor $2$ out of $2 (- 3 x) + 2 (- 4)$ .

$\frac{14}{{(2 (- 3 x - 4))}^{2}}$

Step 1.3.4.2

Apply the product rule to $2 (- 3 x - 4)$ .

$\frac{14}{2^{2} {(- 3 x - 4)}^{2}}$

Step 1.3.4.3

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

$\frac{14}{4 {(- 3 x - 4)}^{2}}$

Step 1.3.5

Cancel the common factor of $14$ and $4$ .

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Step 1.3.5.1

Factor $2$ out of $14$ .

$\frac{2 (7)}{4 {(- 3 x - 4)}^{2}}$

Step 1.3.5.2

Cancel the common factors.

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Step 1.3.5.2.1

Factor $2$ out of $4 {(- 3 x - 4)}^{2}$ .

$\frac{2 (7)}{2 (2 {(- 3 x - 4)}^{2})}$

Step 1.3.5.2.2

Cancel the common factor.

$\frac{2 \cdot 7}{2 (2 {(- 3 x - 4)}^{2})}$

Step 1.3.5.2.3

Rewrite the expression.

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

Step 2

Find the second derivative.

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Step 2.1

Differentiate using the Constant Multiple Rule.

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Step 2.1.1

Since $\frac{7}{2}$ is constant with respect to $x$ , the derivative of $\frac{7}{2 {(- 3 x - 4)}^{2}}$ with respect to $x$ is $\frac{7}{2} \frac{d}{d x} [\frac{1}{{(- 3 x - 4)}^{2}}]$ .

$\frac{7}{2} \frac{d}{d x} [\frac{1}{{(- 3 x - 4)}^{2}}]$

Step 2.1.2

Apply basic rules of exponents.

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Step 2.1.2.1

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

$\frac{7}{2} \frac{d}{d x} [{({(- 3 x - 4)}^{2})}^{- 1}]$

Step 2.1.2.2

Multiply the exponents in ${({(- 3 x - 4)}^{2})}^{- 1}$ .

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Step 2.1.2.2.1

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

$\frac{7}{2} \frac{d}{d x} [{(- 3 x - 4)}^{2 \cdot - 1}]$

Step 2.1.2.2.2

Multiply $2$ by $- 1$ .

$\frac{7}{2} \frac{d}{d x} [{(- 3 x - 4)}^{- 2}]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 2}$ and $g (x) = - 3 x - 4$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $- 3 x - 4$ .

$\frac{7}{2} (\frac{d}{d u} [u^{- 2}] \frac{d}{d x} [- 3 x - 4])$

Step 2.2.2

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

$\frac{7}{2} (- 2 u^{- 3} \frac{d}{d x} [- 3 x - 4])$

Step 2.2.3

Replace all occurrences of $u$ with $- 3 x - 4$ .

$\frac{7}{2} (- 2 {(- 3 x - 4)}^{- 3} \frac{d}{d x} [- 3 x - 4])$

Step 2.3

Differentiate.

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Step 2.3.1

Combine $- 2$ and $\frac{7}{2}$ .

$\frac{- 2 \cdot 7}{2} ({(- 3 x - 4)}^{- 3} \frac{d}{d x} [- 3 x - 4])$

Step 2.3.2

Simplify terms.

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Step 2.3.2.1

Multiply $- 2$ by $7$ .

$\frac{- 14}{2} ({(- 3 x - 4)}^{- 3} \frac{d}{d x} [- 3 x - 4])$

Step 2.3.2.2

Combine ${(- 3 x - 4)}^{- 3}$ and $\frac{- 14}{2}$ .

$\frac{{(- 3 x - 4)}^{- 3} \cdot - 14}{2} \frac{d}{d x} [- 3 x - 4]$

Step 2.3.2.3

Simplify the expression.

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Step 2.3.2.3.1

Move $- 14$ to the left of ${(- 3 x - 4)}^{- 3}$ .

$\frac{- 14 \cdot {(- 3 x - 4)}^{- 3}}{2} \frac{d}{d x} [- 3 x - 4]$

Step 2.3.2.3.2

Move ${(- 3 x - 4)}^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 14}{2 {(- 3 x - 4)}^{3}} \frac{d}{d x} [- 3 x - 4]$

Step 2.3.2.4

Cancel the common factor of $- 14$ and $2$ .

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Step 2.3.2.4.1

Factor $2$ out of $- 14$ .

$\frac{2 \cdot - 7}{2 {(- 3 x - 4)}^{3}} \frac{d}{d x} [- 3 x - 4]$

Step 2.3.2.4.2

Cancel the common factors.

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Step 2.3.2.4.2.1

Factor $2$ out of $2 {(- 3 x - 4)}^{3}$ .

$\frac{2 \cdot - 7}{2 ({(- 3 x - 4)}^{3})} \frac{d}{d x} [- 3 x - 4]$

Step 2.3.2.4.2.2

Cancel the common factor.

$\frac{2 \cdot - 7}{2 {(- 3 x - 4)}^{3}} \frac{d}{d x} [- 3 x - 4]$

Step 2.3.2.4.2.3

Rewrite the expression.

$\frac{- 7}{{(- 3 x - 4)}^{3}} \frac{d}{d x} [- 3 x - 4]$

Step 2.3.2.5

Move the negative in front of the fraction.

$- \frac{7}{{(- 3 x - 4)}^{3}} \frac{d}{d x} [- 3 x - 4]$

Step 2.3.3

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

$- \frac{7}{{(- 3 x - 4)}^{3}} (\frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 4])$

Step 2.3.4

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

$- \frac{7}{{(- 3 x - 4)}^{3}} (- 3 \frac{d}{d x} [x] + \frac{d}{d x} [- 4])$

Step 2.3.5

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

$- \frac{7}{{(- 3 x - 4)}^{3}} (- 3 \cdot 1 + \frac{d}{d x} [- 4])$

Step 2.3.6

Multiply $- 3$ by $1$ .

$- \frac{7}{{(- 3 x - 4)}^{3}} (- 3 + \frac{d}{d x} [- 4])$

Step 2.3.7

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

$- \frac{7}{{(- 3 x - 4)}^{3}} (- 3 + 0)$

Step 2.3.8

Combine fractions.

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Step 2.3.8.1

Add $- 3$ and $0$ .

$- \frac{7}{{(- 3 x - 4)}^{3}} \cdot - 3$

Step 2.3.8.2

Multiply $- 3$ by $- 1$ .

$3 \frac{7}{{(- 3 x - 4)}^{3}}$

Step 2.3.8.3

Combine $3$ and $\frac{7}{{(- 3 x - 4)}^{3}}$ .

$\frac{3 \cdot 7}{{(- 3 x - 4)}^{3}}$

Step 2.3.8.4

Multiply $3$ by $7$ .

$f'' (x) = \frac{21}{{(- 3 x - 4)}^{3}}$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $\frac{21}{{(- 3 x - 4)}^{3}}$ .

$\frac{21}{{(- 3 x - 4)}^{3}}$