Calculus Examples

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

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

Apply basic rules of exponents.

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Step 1.1.1.1

Rewrite $\frac{1}{{(2 x - 5)}^{\frac{1}{3}}}$ as ${({(2 x - 5)}^{\frac{1}{3}})}^{- 1}$ .

$\frac{d}{d x} [{({(2 x - 5)}^{\frac{1}{3}})}^{- 1}]$

Step 1.1.1.2

Multiply the exponents in ${({(2 x - 5)}^{\frac{1}{3}})}^{- 1}$ .

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Step 1.1.1.2.1

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

$\frac{d}{d x} [{(2 x - 5)}^{\frac{1}{3} \cdot - 1}]$

Step 1.1.1.2.2

Combine $\frac{1}{3}$ and $- 1$ .

$\frac{d}{d x} [{(2 x - 5)}^{\frac{- 1}{3}}]$

Step 1.1.1.2.3

Move the negative in front of the fraction.

$\frac{d}{d x} [{(2 x - 5)}^{- \frac{1}{3}}]$

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

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Step 1.1.2.1

To apply the Chain Rule, set $u$ as $2 x - 5$ .

$\frac{d}{d u} [u^{- \frac{1}{3}}] \frac{d}{d x} [2 x - 5]$

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 $n = - \frac{1}{3}$ .

$- \frac{1}{3} u^{- \frac{1}{3} - 1} \frac{d}{d x} [2 x - 5]$

Step 1.1.2.3

Replace all occurrences of $u$ with $2 x - 5$ .

$- \frac{1}{3} {(2 x - 5)}^{- \frac{1}{3} - 1} \frac{d}{d x} [2 x - 5]$

Step 1.1.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{1}{3} {(2 x - 5)}^{- \frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [2 x - 5]$

Step 1.1.4

Combine $- 1$ and $\frac{3}{3}$ .

$- \frac{1}{3} {(2 x - 5)}^{- \frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [2 x - 5]$

Step 1.1.5

Combine the numerators over the common denominator.

$- \frac{1}{3} {(2 x - 5)}^{\frac{- 1 - 1 \cdot 3}{3}} \frac{d}{d x} [2 x - 5]$

Step 1.1.6

Simplify the numerator.

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Step 1.1.6.1

Multiply $- 1$ by $3$ .

$- \frac{1}{3} {(2 x - 5)}^{\frac{- 1 - 3}{3}} \frac{d}{d x} [2 x - 5]$

Step 1.1.6.2

Subtract $3$ from $- 1$ .

$- \frac{1}{3} {(2 x - 5)}^{\frac{- 4}{3}} \frac{d}{d x} [2 x - 5]$

Step 1.1.7

Combine fractions.

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Step 1.1.7.1

Move the negative in front of the fraction.

$- \frac{1}{3} {(2 x - 5)}^{- \frac{4}{3}} \frac{d}{d x} [2 x - 5]$

Step 1.1.7.2

Combine ${(2 x - 5)}^{- \frac{4}{3}}$ and $\frac{1}{3}$ .

$- \frac{{(2 x - 5)}^{- \frac{4}{3}}}{3} \frac{d}{d x} [2 x - 5]$

Step 1.1.7.3

Move ${(2 x - 5)}^{- \frac{4}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{3 {(2 x - 5)}^{\frac{4}{3}}} \frac{d}{d x} [2 x - 5]$

Step 1.1.8

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

$- \frac{1}{3 {(2 x - 5)}^{\frac{4}{3}}} (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 5])$

Step 1.1.9

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

$- \frac{1}{3 {(2 x - 5)}^{\frac{4}{3}}} (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 5])$

Step 1.1.10

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

$- \frac{1}{3 {(2 x - 5)}^{\frac{4}{3}}} (2 \cdot 1 + \frac{d}{d x} [- 5])$

Step 1.1.11

Multiply $2$ by $1$ .

$- \frac{1}{3 {(2 x - 5)}^{\frac{4}{3}}} (2 + \frac{d}{d x} [- 5])$

Step 1.1.12

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

$- \frac{1}{3 {(2 x - 5)}^{\frac{4}{3}}} (2 + 0)$

Step 1.1.13

Combine fractions.

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Step 1.1.13.1

Add $2$ and $0$ .

$- \frac{1}{3 {(2 x - 5)}^{\frac{4}{3}}} \cdot 2$

Step 1.1.13.2

Multiply $2$ by $- 1$ .

$- 2 \frac{1}{3 {(2 x - 5)}^{\frac{4}{3}}}$

Step 1.1.13.3

Combine $- 2$ and $\frac{1}{3 {(2 x - 5)}^{\frac{4}{3}}}$ .

$\frac{- 2}{3 {(2 x - 5)}^{\frac{4}{3}}}$

Step 1.1.13.4

Move the negative in front of the fraction.

$f' (x) = - \frac{2}{3 {(2 x - 5)}^{\frac{4}{3}}}$

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 $- \frac{2}{3}$ is constant with respect to $x$ , the derivative of $- \frac{2}{3 {(2 x - 5)}^{\frac{4}{3}}}$ with respect to $x$ is $- \frac{2}{3} \frac{d}{d x} [\frac{1}{{(2 x - 5)}^{\frac{4}{3}}}]$ .

$- \frac{2}{3} \frac{d}{d x} [\frac{1}{{(2 x - 5)}^{\frac{4}{3}}}]$

Step 1.2.1.2

Apply basic rules of exponents.

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Step 1.2.1.2.1

Rewrite $\frac{1}{{(2 x - 5)}^{\frac{4}{3}}}$ as ${({(2 x - 5)}^{\frac{4}{3}})}^{- 1}$ .

$- \frac{2}{3} \frac{d}{d x} [{({(2 x - 5)}^{\frac{4}{3}})}^{- 1}]$

Step 1.2.1.2.2

Multiply the exponents in ${({(2 x - 5)}^{\frac{4}{3}})}^{- 1}$ .

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Step 1.2.1.2.2.1

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

$- \frac{2}{3} \frac{d}{d x} [{(2 x - 5)}^{\frac{4}{3} \cdot - 1}]$

Step 1.2.1.2.2.2

Multiply $\frac{4}{3} \cdot - 1$ .

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Step 1.2.1.2.2.2.1

Combine $\frac{4}{3}$ and $- 1$ .

$- \frac{2}{3} \frac{d}{d x} [{(2 x - 5)}^{\frac{4 \cdot - 1}{3}}]$

Step 1.2.1.2.2.2.2

Multiply $4$ by $- 1$ .

$- \frac{2}{3} \frac{d}{d x} [{(2 x - 5)}^{\frac{- 4}{3}}]$

Step 1.2.1.2.2.3

Move the negative in front of the fraction.

$- \frac{2}{3} \frac{d}{d x} [{(2 x - 5)}^{- \frac{4}{3}}]$

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 $f (x) = x^{- \frac{4}{3}}$ and $g (x) = 2 x - 5$ .

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Step 1.2.2.1

To apply the Chain Rule, set $u$ as $2 x - 5$ .

$- \frac{2}{3} (\frac{d}{d u} [u^{- \frac{4}{3}}] \frac{d}{d x} [2 x - 5])$

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 $n = - \frac{4}{3}$ .

$- \frac{2}{3} (- \frac{4}{3} u^{- \frac{4}{3} - 1} \frac{d}{d x} [2 x - 5])$

Step 1.2.2.3

Replace all occurrences of $u$ with $2 x - 5$ .

$- \frac{2}{3} (- \frac{4}{3} {(2 x - 5)}^{- \frac{4}{3} - 1} \frac{d}{d x} [2 x - 5])$

Step 1.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{2}{3} (- \frac{4}{3} {(2 x - 5)}^{- \frac{4}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [2 x - 5])$

Step 1.2.4

Combine $- 1$ and $\frac{3}{3}$ .

$- \frac{2}{3} (- \frac{4}{3} {(2 x - 5)}^{- \frac{4}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [2 x - 5])$

Step 1.2.5

Combine the numerators over the common denominator.

$- \frac{2}{3} (- \frac{4}{3} {(2 x - 5)}^{\frac{- 4 - 1 \cdot 3}{3}} \frac{d}{d x} [2 x - 5])$

Step 1.2.6

Simplify the numerator.

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Step 1.2.6.1

Multiply $- 1$ by $3$ .

$- \frac{2}{3} (- \frac{4}{3} {(2 x - 5)}^{\frac{- 4 - 3}{3}} \frac{d}{d x} [2 x - 5])$

Step 1.2.6.2

Subtract $3$ from $- 4$ .

$- \frac{2}{3} (- \frac{4}{3} {(2 x - 5)}^{\frac{- 7}{3}} \frac{d}{d x} [2 x - 5])$

Step 1.2.7

Combine fractions.

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Step 1.2.7.1

Move the negative in front of the fraction.

$- \frac{2}{3} (- \frac{4}{3} {(2 x - 5)}^{- \frac{7}{3}} \frac{d}{d x} [2 x - 5])$

Step 1.2.7.2

Combine ${(2 x - 5)}^{- \frac{7}{3}}$ and $\frac{4}{3}$ .

$- \frac{2}{3} (- \frac{{(2 x - 5)}^{- \frac{7}{3}} \cdot 4}{3} \frac{d}{d x} [2 x - 5])$

Step 1.2.7.3

Simplify the expression.

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Step 1.2.7.3.1

Move $4$ to the left of ${(2 x - 5)}^{- \frac{7}{3}}$ .

$- \frac{2}{3} (- \frac{4 \cdot {(2 x - 5)}^{- \frac{7}{3}}}{3} \frac{d}{d x} [2 x - 5])$

Step 1.2.7.3.2

Move ${(2 x - 5)}^{- \frac{7}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{2}{3} (- \frac{4}{3 {(2 x - 5)}^{\frac{7}{3}}} \frac{d}{d x} [2 x - 5])$

Step 1.2.7.3.3

Multiply $- 1$ by $- 1$ .

$1 (\frac{2}{3}) (\frac{4}{3 {(2 x - 5)}^{\frac{7}{3}}} \frac{d}{d x} [2 x - 5])$

Step 1.2.7.3.4

Multiply $\frac{2}{3}$ by $1$ .

$\frac{2}{3} (\frac{4}{3 {(2 x - 5)}^{\frac{7}{3}}} \frac{d}{d x} [2 x - 5])$

Step 1.2.7.4

Multiply $\frac{4}{3 {(2 x - 5)}^{\frac{7}{3}}}$ by $\frac{2}{3}$ .

$\frac{4 \cdot 2}{3 {(2 x - 5)}^{\frac{7}{3}} \cdot 3} \frac{d}{d x} [2 x - 5]$

Step 1.2.7.5

Multiply.

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Step 1.2.7.5.1

Multiply $4$ by $2$ .

$\frac{8}{3 {(2 x - 5)}^{\frac{7}{3}} \cdot 3} \frac{d}{d x} [2 x - 5]$

Step 1.2.7.5.2

Multiply $3$ by $3$ .

$\frac{8}{9 {(2 x - 5)}^{\frac{7}{3}}} \frac{d}{d x} [2 x - 5]$

Step 1.2.8

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

$\frac{8}{9 {(2 x - 5)}^{\frac{7}{3}}} (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 5])$

Step 1.2.9

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

$\frac{8}{9 {(2 x - 5)}^{\frac{7}{3}}} (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 5])$

Step 1.2.10

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

$\frac{8}{9 {(2 x - 5)}^{\frac{7}{3}}} (2 \cdot 1 + \frac{d}{d x} [- 5])$

Step 1.2.11

Multiply $2$ by $1$ .

$\frac{8}{9 {(2 x - 5)}^{\frac{7}{3}}} (2 + \frac{d}{d x} [- 5])$

Step 1.2.12

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

$\frac{8}{9 {(2 x - 5)}^{\frac{7}{3}}} (2 + 0)$

Step 1.2.13

Combine fractions.

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Step 1.2.13.1

Add $2$ and $0$ .

$\frac{8}{9 {(2 x - 5)}^{\frac{7}{3}}} \cdot 2$

Step 1.2.13.2

Combine $\frac{8}{9 {(2 x - 5)}^{\frac{7}{3}}}$ and $2$ .

$\frac{8 \cdot 2}{9 {(2 x - 5)}^{\frac{7}{3}}}$

Step 1.2.13.3

Multiply $8$ by $2$ .

$f'' (x) = \frac{16}{9 {(2 x - 5)}^{\frac{7}{3}}}$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{16}{9 {(2 x - 5)}^{\frac{7}{3}}}$ .

$\frac{16}{9 {(2 x - 5)}^{\frac{7}{3}}}$

Step 2

Set the second derivative equal to $0$ then solve the equation $\frac{16}{9 {(2 x - 5)}^{\frac{7}{3}}} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$\frac{16}{9 {(2 x - 5)}^{\frac{7}{3}}} = 0$

Step 2.2

Set the numerator equal to zero.

$16 = 0$

Step 2.3

Since $16 \neq 0$ , there are no solutions.

No solution

Step 3

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

No Inflection Points