Calculus Examples

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

Step 1

Find the second 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

Since $6$ is constant with respect to $x$ , the derivative of $\frac{6}{x - 5}$ with respect to $x$ is $6 \frac{d}{d x} [\frac{1}{x - 5}]$ .

$6 \frac{d}{d x} [\frac{1}{x - 5}]$

Step 1.1.1.2

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

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

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 $g (x) = x - 5$ .

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Step 1.1.2.1

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

$6 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x - 5])$

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 $n = - 1$ .

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

Step 1.1.2.3

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

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

Step 1.1.3

Differentiate.

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Step 1.1.3.1

Multiply $- 1$ by $6$ .

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

Step 1.1.3.2

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

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

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 $n = 1$ .

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

Step 1.1.3.4

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

$- 6 {(x - 5)}^{- 2} (1 + 0)$

Step 1.1.3.5

Simplify the expression.

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Step 1.1.3.5.1

Add $1$ and $0$ .

$- 6 {(x - 5)}^{- 2} \cdot 1$

Step 1.1.3.5.2

Multiply $- 6$ by $1$ .

$- 6 {(x - 5)}^{- 2}$

Step 1.1.4

Simplify.

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Step 1.1.4.1

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

$- 6 \frac{1}{{(x - 5)}^{2}}$

Step 1.1.4.2

Combine terms.

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Step 1.1.4.2.1

Combine $- 6$ and $\frac{1}{{(x - 5)}^{2}}$ .

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

Step 1.1.4.2.2

Move the negative in front of the fraction.

$f' (x) = - \frac{6}{{(x - 5)}^{2}}$

Step 1.2

Find the second derivative.

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Step 1.2.1

Differentiate using the Constant Multiple Rule.

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Step 1.2.1.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- \frac{6}{{(x - 5)}^{2}}$ with respect to $x$ is $- 6 \frac{d}{d x} [\frac{1}{{(x - 5)}^{2}}]$ .

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

Step 1.2.1.2

Apply basic rules of exponents.

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Step 1.2.1.2.1

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

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

Step 1.2.1.2.2

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

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Step 1.2.1.2.2.1

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

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

Step 1.2.1.2.2.2

Multiply $2$ by $- 1$ .

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

Step 1.2.2

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 $g (x) = x - 5$ .

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Step 1.2.2.1

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

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

Step 1.2.2.2

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

$- 6 (- 2 u^{- 3} \frac{d}{d x} [x - 5])$

Step 1.2.2.3

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

$- 6 (- 2 {(x - 5)}^{- 3} \frac{d}{d x} [x - 5])$

Step 1.2.3

Differentiate.

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Step 1.2.3.1

Multiply $- 2$ by $- 6$ .

$12 ({(x - 5)}^{- 3} \frac{d}{d x} [x - 5])$

Step 1.2.3.2

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

$12 {(x - 5)}^{- 3} (\frac{d}{d x} [x] + \frac{d}{d x} [- 5])$

Step 1.2.3.3

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

$12 {(x - 5)}^{- 3} (1 + \frac{d}{d x} [- 5])$

Step 1.2.3.4

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

$12 {(x - 5)}^{- 3} (1 + 0)$

Step 1.2.3.5

Simplify the expression.

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Step 1.2.3.5.1

Add $1$ and $0$ .

$12 {(x - 5)}^{- 3} \cdot 1$

Step 1.2.3.5.2

Multiply $12$ by $1$ .

$12 {(x - 5)}^{- 3}$

Step 1.2.4

Simplify.

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Step 1.2.4.1

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

$12 \frac{1}{{(x - 5)}^{3}}$

Step 1.2.4.2

Combine $12$ and $\frac{1}{{(x - 5)}^{3}}$ .

$f'' (x) = \frac{12}{{(x - 5)}^{3}}$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{12}{{(x - 5)}^{3}}$ .

$\frac{12}{{(x - 5)}^{3}}$

Step 2

Set the second derivative equal to $0$ then solve the equation $\frac{12}{{(x - 5)}^{3}} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$\frac{12}{{(x - 5)}^{3}} = 0$

Step 2.2

Set the numerator equal to zero.

$12 = 0$

Step 2.3

Since $12 \neq 0$ , there are no solutions.

No solution

Step 3

No values found that can make the second derivative equal to $0$ .

No Inflection Points