Calculus Examples

$f (x) = \frac{1 - 8 x}{8 - 6 x}$

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 $f (x) = 1 - 8 x$ and $g (x) = 8 - 6 x$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $1 - 8 x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 8 x]$ .

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

Step 1.2.2

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

$\frac{(8 - 6 x) (0 + \frac{d}{d x} [- 8 x]) - (1 - 8 x) \frac{d}{d x} [8 - 6 x]}{{(8 - 6 x)}^{2}}$

Step 1.2.3

Add $0$ and $\frac{d}{d x} [- 8 x]$ .

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

Step 1.2.4

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x$ with respect to $x$ is $- 8 \frac{d}{d x} [x]$ .

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

Step 1.2.5

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

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

Step 1.2.6

Simplify the expression.

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Step 1.2.6.1

Multiply $- 8$ by $1$ .

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

Step 1.2.6.2

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

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

Step 1.2.7

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

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

Step 1.2.8

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

$\frac{- 8 (8 - 6 x) - (1 - 8 x) (0 + \frac{d}{d x} [- 6 x])}{{(8 - 6 x)}^{2}}$

Step 1.2.9

Add $0$ and $\frac{d}{d x} [- 6 x]$ .

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

Step 1.2.10

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

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

Step 1.2.11

Multiply $- 6$ by $- 1$ .

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

Step 1.2.12

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

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

Step 1.2.13

Multiply $6$ by $1$ .

$\frac{- 8 (8 - 6 x) + 6 (1 - 8 x)}{{(8 - 6 x)}^{2}}$

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Apply the distributive property.

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

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 $- 8$ by $8$ .

$\frac{- 64 - 8 (- 6 x) + 6 \cdot 1 + 6 (- 8 x)}{{(8 - 6 x)}^{2}}$

Step 1.3.3.1.2

Multiply $- 6$ by $- 8$ .

$\frac{- 64 + 48 x + 6 \cdot 1 + 6 (- 8 x)}{{(8 - 6 x)}^{2}}$

Step 1.3.3.1.3

Multiply $6$ by $1$ .

$\frac{- 64 + 48 x + 6 + 6 (- 8 x)}{{(8 - 6 x)}^{2}}$

Step 1.3.3.1.4

Multiply $- 8$ by $6$ .

$\frac{- 64 + 48 x + 6 - 48 x}{{(8 - 6 x)}^{2}}$

Step 1.3.3.2

Combine the opposite terms in $- 64 + 48 x + 6 - 48 x$ .

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Step 1.3.3.2.1

Subtract $48 x$ from $48 x$ .

$\frac{- 64 + 0 + 6}{{(8 - 6 x)}^{2}}$

Step 1.3.3.2.2

Add $- 64$ and $0$ .

$\frac{- 64 + 6}{{(8 - 6 x)}^{2}}$

Step 1.3.3.3

Add $- 64$ and $6$ .

$\frac{- 58}{{(8 - 6 x)}^{2}}$

Step 1.3.4

Move the negative in front of the fraction.

$- \frac{58}{{(8 - 6 x)}^{2}}$

Step 1.3.5

Simplify the denominator.

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Step 1.3.5.1

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

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Step 1.3.5.1.1

Factor $2$ out of $8$ .

$- \frac{58}{{(2 (4) - 6 x)}^{2}}$

Step 1.3.5.1.2

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

$- \frac{58}{{(2 (4) + 2 (- 3 x))}^{2}}$

Step 1.3.5.1.3

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

$- \frac{58}{{(2 (4 - 3 x))}^{2}}$

Step 1.3.5.2

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

$- \frac{58}{2^{2} {(4 - 3 x)}^{2}}$

Step 1.3.5.3

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

$- \frac{58}{4 {(4 - 3 x)}^{2}}$

Step 1.3.6

Cancel the common factor of $58$ and $4$ .

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Step 1.3.6.1

Factor $2$ out of $58$ .

$- \frac{2 (29)}{4 {(4 - 3 x)}^{2}}$

Step 1.3.6.2

Cancel the common factors.

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Step 1.3.6.2.1

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

$- \frac{2 (29)}{2 (2 {(4 - 3 x)}^{2})}$

Step 1.3.6.2.2

Cancel the common factor.

$- \frac{2 \cdot 29}{2 (2 {(4 - 3 x)}^{2})}$

Step 1.3.6.2.3

Rewrite the expression.

$f' (x) = - \frac{29}{2 {(4 - 3 x)}^{2}}$

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 $- \frac{29}{2}$ is constant with respect to $x$ , the derivative of $- \frac{29}{2 {(4 - 3 x)}^{2}}$ with respect to $x$ is $- \frac{29}{2} \frac{d}{d x} [\frac{1}{{(4 - 3 x)}^{2}}]$ .

$- \frac{29}{2} \frac{d}{d x} [\frac{1}{{(4 - 3 x)}^{2}}]$

Step 2.1.2

Apply basic rules of exponents.

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Step 2.1.2.1

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

$- \frac{29}{2} \frac{d}{d x} [{({(4 - 3 x)}^{2})}^{- 1}]$

Step 2.1.2.2

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

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Step 2.1.2.2.1

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

$- \frac{29}{2} \frac{d}{d x} [{(4 - 3 x)}^{2 \cdot - 1}]$

Step 2.1.2.2.2

Multiply $2$ by $- 1$ .

$- \frac{29}{2} \frac{d}{d x} [{(4 - 3 x)}^{- 2}]$

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 $g (x) = 4 - 3 x$ .

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Step 2.2.1

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

$- \frac{29}{2} (\frac{d}{d u} [u^{- 2}] \frac{d}{d x} [4 - 3 x])$

Step 2.2.2

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

$- \frac{29}{2} (- 2 u^{- 3} \frac{d}{d x} [4 - 3 x])$

Step 2.2.3

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

$- \frac{29}{2} (- 2 {(4 - 3 x)}^{- 3} \frac{d}{d x} [4 - 3 x])$

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $- 2$ by $- 1$ .

$2 (\frac{29}{2}) ({(4 - 3 x)}^{- 3} \frac{d}{d x} [4 - 3 x])$

Step 2.3.2

Simplify terms.

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Step 2.3.2.1

Combine $2$ and $\frac{29}{2}$ .

$\frac{2 \cdot 29}{2} ({(4 - 3 x)}^{- 3} \frac{d}{d x} [4 - 3 x])$

Step 2.3.2.2

Multiply $2$ by $29$ .

$\frac{58}{2} ({(4 - 3 x)}^{- 3} \frac{d}{d x} [4 - 3 x])$

Step 2.3.2.3

Combine ${(4 - 3 x)}^{- 3}$ and $\frac{58}{2}$ .

$\frac{{(4 - 3 x)}^{- 3} \cdot 58}{2} \frac{d}{d x} [4 - 3 x]$

Step 2.3.2.4

Simplify the expression.

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Step 2.3.2.4.1

Move $58$ to the left of ${(4 - 3 x)}^{- 3}$ .

$\frac{58 \cdot {(4 - 3 x)}^{- 3}}{2} \frac{d}{d x} [4 - 3 x]$

Step 2.3.2.4.2

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

$\frac{58}{2 {(4 - 3 x)}^{3}} \frac{d}{d x} [4 - 3 x]$

Step 2.3.2.5

Cancel the common factor of $58$ and $2$ .

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Step 2.3.2.5.1

Factor $2$ out of $58$ .

$\frac{2 \cdot 29}{2 {(4 - 3 x)}^{3}} \frac{d}{d x} [4 - 3 x]$

Step 2.3.2.5.2

Cancel the common factors.

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Step 2.3.2.5.2.1

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

$\frac{2 \cdot 29}{2 ({(4 - 3 x)}^{3})} \frac{d}{d x} [4 - 3 x]$

Step 2.3.2.5.2.2

Cancel the common factor.

$\frac{2 \cdot 29}{2 {(4 - 3 x)}^{3}} \frac{d}{d x} [4 - 3 x]$

Step 2.3.2.5.2.3

Rewrite the expression.

$\frac{29}{{(4 - 3 x)}^{3}} \frac{d}{d x} [4 - 3 x]$

Step 2.3.3

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

$\frac{29}{{(4 - 3 x)}^{3}} (\frac{d}{d x} [4] + \frac{d}{d x} [- 3 x])$

Step 2.3.4

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

$\frac{29}{{(4 - 3 x)}^{3}} (0 + \frac{d}{d x} [- 3 x])$

Step 2.3.5

Add $0$ and $\frac{d}{d x} [- 3 x]$ .

$\frac{29}{{(4 - 3 x)}^{3}} \frac{d}{d x} [- 3 x]$

Step 2.3.6

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

$\frac{29}{{(4 - 3 x)}^{3}} (- 3 \frac{d}{d x} [x])$

Step 2.3.7

Combine fractions.

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Step 2.3.7.1

Combine $- 3$ and $\frac{29}{{(4 - 3 x)}^{3}}$ .

$\frac{- 3 \cdot 29}{{(4 - 3 x)}^{3}} \frac{d}{d x} [x]$

Step 2.3.7.2

Simplify the expression.

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Step 2.3.7.2.1

Multiply $- 3$ by $29$ .

$\frac{- 87}{{(4 - 3 x)}^{3}} \frac{d}{d x} [x]$

Step 2.3.7.2.2

Move the negative in front of the fraction.

$- \frac{87}{{(4 - 3 x)}^{3}} \frac{d}{d x} [x]$

Step 2.3.8

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

$- \frac{87}{{(4 - 3 x)}^{3}} \cdot 1$

Step 2.3.9

Multiply $- 1$ by $1$ .

$f'' (x) = - \frac{87}{{(4 - 3 x)}^{3}}$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{87}{{(4 - 3 x)}^{3}}$ .

$- \frac{87}{{(4 - 3 x)}^{3}}$