Calculus Examples

f (x) = x + 3 {(x - 1)}^{\frac{1}{3}}

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

Step 1

Find the second derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

Differentiate.

Tap for more steps...

Step 1.1.1.1

By the Sum Rule, the derivative of

$x + 3 {(x - 1)}^{\frac{1}{3}}$ with respect to

$x$ is

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

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

Step 1.1.1.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Step 1.1.2

Evaluate

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

Tap for more steps...

Step 1.1.2.1

Since

$3$ is constant with respect to

$x$ , the derivative of

$3 {(x - 1)}^{\frac{1}{3}}$ with respect to

$x$ is

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

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

Step 1.1.2.2

Differentiate using the chain rule, which states that

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

$f (x) = x^{\frac{1}{3}}$ and

$g (x) = x - 1$ .

Tap for more steps...

Step 1.1.2.2.1

To apply the Chain Rule, set

$u$ as

$x - 1$ .

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

Step 1.1.2.2.2

Differentiate using the Power Rule which states that

$\frac{d}{d u} [u^{n}]$ is

$n u^{n - 1}$ where

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

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

Step 1.1.2.2.3

Replace all occurrences of

$u$ with

$x - 1$ .

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

Step 1.1.2.3

By the Sum Rule, the derivative of

$x - 1$ with respect to

$x$ is

$\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

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

Step 1.1.2.4

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Step 1.1.2.5

Since

$- 1$ is constant with respect to

$x$ , the derivative of

$- 1$ with respect to

$x$ is

$0$ .

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

Step 1.1.2.6

To write

$- 1$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

$1 + 3 (\frac{1}{3} {(x - 1)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} (1 + 0))$

Step 1.1.2.7

Combine

$- 1$ and

$\frac{3}{3}$ .

$1 + 3 (\frac{1}{3} {(x - 1)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} (1 + 0))$

Step 1.1.2.8

Combine the numerators over the common denominator.

$1 + 3 (\frac{1}{3} {(x - 1)}^{\frac{1 - 1 \cdot 3}{3}} (1 + 0))$

Step 1.1.2.9

Simplify the numerator.

Tap for more steps...

Step 1.1.2.9.1

Multiply

$- 1$ by

$3$ .

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

Step 1.1.2.9.2

Subtract

$3$ from

$1$ .

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

Step 1.1.2.10

Move the negative in front of the fraction.

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

Step 1.1.2.11

Add

$1$ and

$0$ .

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

Step 1.1.2.12

Combine

$\frac{1}{3}$ and

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

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

Step 1.1.2.13

Multiply

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

$1$ .

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

Step 1.1.2.14

Move

${(x - 1)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.1.2.15

Combine

$3$ and

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

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

Step 1.1.2.16

Cancel the common factor.

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

Step 1.1.2.17

Rewrite the expression.

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

Step 1.2

Find the second derivative.

Tap for more steps...

Step 1.2.1

Differentiate.

Tap for more steps...

Step 1.2.1.1

By the Sum Rule, the derivative of

$1 + \frac{1}{{(x - 1)}^{\frac{2}{3}}}$ with respect to

$x$ is

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

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

Step 1.2.1.2

Since

$1$ is constant with respect to

$x$ , the derivative of

$1$ with respect to

$x$ is

$0$ .

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

Step 1.2.2

Evaluate

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

Tap for more steps...

Step 1.2.2.1

Rewrite

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

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

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

Step 1.2.2.2

Differentiate using the chain rule, which states that

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

$f (x) = x^{- 1}$ and

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

Tap for more steps...

Step 1.2.2.2.1

To apply the Chain Rule, set

$u_{1}$ as

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

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

Step 1.2.2.2.2

Differentiate using the Power Rule which states that

$\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is

$n {u_{1}}^{n - 1}$ where

$n = - 1$ .

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

Step 1.2.2.2.3

Replace all occurrences of

$u_{1}$ with

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

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

Step 1.2.2.3

Differentiate using the chain rule, which states that

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

$f (x) = x^{\frac{2}{3}}$ and

$g (x) = x - 1$ .

Tap for more steps...

Step 1.2.2.3.1

To apply the Chain Rule, set

$u_{2}$ as

$x - 1$ .

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

Step 1.2.2.3.2

Differentiate using the Power Rule which states that

$\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is

$n {u_{2}}^{n - 1}$ where

$n = \frac{2}{3}$ .

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

Step 1.2.2.3.3

Replace all occurrences of

$u_{2}$ with

$x - 1$ .

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

Step 1.2.2.4

By the Sum Rule, the derivative of

$x - 1$ with respect to

$x$ is

$\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

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

Step 1.2.2.5

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Step 1.2.2.6

Since

$- 1$ is constant with respect to

$x$ , the derivative of

$- 1$ with respect to

$x$ is

$0$ .

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

Step 1.2.2.7

Multiply the exponents in

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

Tap for more steps...

Step 1.2.2.7.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.2.2.7.2

Multiply

$\frac{2}{3} \cdot - 2$ .

Tap for more steps...

Step 1.2.2.7.2.1

Combine

$\frac{2}{3}$ and

$- 2$ .

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

Step 1.2.2.7.2.2

Multiply

$2$ by

$- 2$ .

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

Step 1.2.2.7.3

Move the negative in front of the fraction.

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

Step 1.2.2.8

To write

$- 1$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

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

Step 1.2.2.9

Combine

$- 1$ and

$\frac{3}{3}$ .

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

Step 1.2.2.10

Combine the numerators over the common denominator.

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

Step 1.2.2.11

Simplify the numerator.

Tap for more steps...

Step 1.2.2.11.1

Multiply

$- 1$ by

$3$ .

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

Step 1.2.2.11.2

Subtract

$3$ from

$2$ .

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

Step 1.2.2.12

Move the negative in front of the fraction.

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

Step 1.2.2.13

Add

$1$ and

$0$ .

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

Step 1.2.2.14

Combine

$\frac{2}{3}$ and

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

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

Step 1.2.2.15

Multiply

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

$1$ .

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

Step 1.2.2.16

Move

${(x - 1)}^{- \frac{1}{3}}$ to the denominator using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.2.2.17

Combine

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

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

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

Step 1.2.2.18

Move

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

$b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.2.2.19

Multiply

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

${(x - 1)}^{\frac{4}{3}}$ by adding the exponents.

Tap for more steps...

Step 1.2.2.19.1

Move

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

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

Step 1.2.2.19.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.2.2.19.3

Combine the numerators over the common denominator.

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

Step 1.2.2.19.4

Add

$4$ and

$1$ .

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

Step 1.2.3

Subtract

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

$0$ .

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

Step 1.3

The second derivative of

$f (x)$ with respect to

$x$ is

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

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

Step 2

Set the second derivative equal to

$0$ then solve the equation

$- \frac{2}{3 {(x - 1)}^{\frac{5}{3}}} = 0$ .

Tap for more steps...

Step 2.1

Set the second derivative equal to

$0$ .

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

Step 2.2

Set the numerator equal to zero.

$2 = 0$

Step 2.3

Since

$2 \neq 0$ , there are no solutions.

No solution

Step 3

No values found that can make the second derivative equal to

$0$ .

No Inflection Points