Calculus Examples

$y = {(x^{9} + x)}^{\frac{5}{6}}$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{5}{6}}$ and $g (x) = x^{9} + x$ .

Tap for more steps...

Step 1.1.1

To apply the Chain Rule, set $u$ as $x^{9} + x$ .

$\frac{d}{d u} [u^{\frac{5}{6}}] \frac{d}{d x} [x^{9} + x]$

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{5}{6}$ .

$\frac{5}{6} u^{\frac{5}{6} - 1} \frac{d}{d x} [x^{9} + x]$

Step 1.1.3

Replace all occurrences of $u$ with $x^{9} + x$ .

$\frac{5}{6} {(x^{9} + x)}^{\frac{5}{6} - 1} \frac{d}{d x} [x^{9} + x]$

Step 1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{5}{6} {(x^{9} + x)}^{\frac{5}{6} - 1 \cdot \frac{6}{6}} \frac{d}{d x} [x^{9} + x]$

Step 1.3

Combine $- 1$ and $\frac{6}{6}$ .

$\frac{5}{6} {(x^{9} + x)}^{\frac{5}{6} + \frac{- 1 \cdot 6}{6}} \frac{d}{d x} [x^{9} + x]$

Step 1.4

Combine the numerators over the common denominator.

$\frac{5}{6} {(x^{9} + x)}^{\frac{5 - 1 \cdot 6}{6}} \frac{d}{d x} [x^{9} + x]$

Step 1.5

Simplify the numerator.

Tap for more steps...

Step 1.5.1

Multiply $- 1$ by $6$ .

$\frac{5}{6} {(x^{9} + x)}^{\frac{5 - 6}{6}} \frac{d}{d x} [x^{9} + x]$

Step 1.5.2

Subtract $6$ from $5$ .

$\frac{5}{6} {(x^{9} + x)}^{\frac{- 1}{6}} \frac{d}{d x} [x^{9} + x]$

Step 1.6

Combine fractions.

Tap for more steps...

Step 1.6.1

Move the negative in front of the fraction.

$\frac{5}{6} {(x^{9} + x)}^{- \frac{1}{6}} \frac{d}{d x} [x^{9} + x]$

Step 1.6.2

Combine $\frac{5}{6}$ and ${(x^{9} + x)}^{- \frac{1}{6}}$ .

$\frac{5 {(x^{9} + x)}^{- \frac{1}{6}}}{6} \frac{d}{d x} [x^{9} + x]$

Step 1.6.3

Move ${(x^{9} + x)}^{- \frac{1}{6}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{5}{6 {(x^{9} + x)}^{\frac{1}{6}}} \frac{d}{d x} [x^{9} + x]$

Step 1.7

By the Sum Rule, the derivative of $x^{9} + x$ with respect to $x$ is $\frac{d}{d x} [x^{9}] + \frac{d}{d x} [x]$ .

$\frac{5}{6 {(x^{9} + x)}^{\frac{1}{6}}} (\frac{d}{d x} [x^{9}] + \frac{d}{d x} [x])$

Step 1.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 9$ .

$\frac{5}{6 {(x^{9} + x)}^{\frac{1}{6}}} (9 x^{8} + \frac{d}{d x} [x])$

Step 1.9

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

$\frac{5}{6 {(x^{9} + x)}^{\frac{1}{6}}} (9 x^{8} + 1)$

Step 1.10

Simplify.

Tap for more steps...

Step 1.10.1

Reorder the factors of $\frac{5}{6 {(x^{9} + x)}^{\frac{1}{6}}} (9 x^{8} + 1)$ .

$(9 x^{8} + 1) \frac{5}{6 {(x^{9} + x)}^{\frac{1}{6}}}$

Step 1.10.2

Multiply $9 x^{8} + 1$ by $\frac{5}{6 {(x^{9} + x)}^{\frac{1}{6}}}$ .

$\frac{(9 x^{8} + 1) \cdot 5}{6 {(x^{9} + x)}^{\frac{1}{6}}}$

Step 1.10.3

Move $5$ to the left of $9 x^{8} + 1$ .

$f' (x) = \frac{5 (9 x^{8} + 1)}{6 {(x^{9} + x)}^{\frac{1}{6}}}$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

Since $\frac{5}{6}$ is constant with respect to $x$ , the derivative of $\frac{5 (9 x^{8} + 1)}{6 {(x^{9} + x)}^{\frac{1}{6}}}$ with respect to $x$ is $\frac{5}{6} \frac{d}{d x} [\frac{9 x^{8} + 1}{{(x^{9} + x)}^{\frac{1}{6}}}]$ .

$\frac{5}{6} \frac{d}{d x} [\frac{9 x^{8} + 1}{{(x^{9} + x)}^{\frac{1}{6}}}]$

Step 2.2

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 $f (x) = 9 x^{8} + 1$ and $g (x) = {(x^{9} + x)}^{\frac{1}{6}}$ .

$\frac{5}{6} \cdot \frac{{(x^{9} + x)}^{\frac{1}{6}} \frac{d}{d x} [9 x^{8} + 1] - (9 x^{8} + 1) \frac{d}{d x} [{(x^{9} + x)}^{\frac{1}{6}}]}{{({(x^{9} + x)}^{\frac{1}{6}})}^{2}}$

Step 2.3

Differentiate.

Tap for more steps...

Step 2.3.1

Multiply the exponents in ${({(x^{9} + x)}^{\frac{1}{6}})}^{2}$ .

Tap for more steps...

Step 2.3.1.1

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

$\frac{5}{6} \cdot \frac{{(x^{9} + x)}^{\frac{1}{6}} \frac{d}{d x} [9 x^{8} + 1] - (9 x^{8} + 1) \frac{d}{d x} [{(x^{9} + x)}^{\frac{1}{6}}]}{{(x^{9} + x)}^{\frac{1}{6} \cdot 2}}$

Step 2.3.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.1.2.1

Factor $2$ out of $6$ .

$\frac{5}{6} \cdot \frac{{(x^{9} + x)}^{\frac{1}{6}} \frac{d}{d x} [9 x^{8} + 1] - (9 x^{8} + 1) \frac{d}{d x} [{(x^{9} + x)}^{\frac{1}{6}}]}{{(x^{9} + x)}^{\frac{1}{2 (3)} \cdot 2}}$

Step 2.3.1.2.2

Cancel the common factor.

$\frac{5}{6} \cdot \frac{{(x^{9} + x)}^{\frac{1}{6}} \frac{d}{d x} [9 x^{8} + 1] - (9 x^{8} + 1) \frac{d}{d x} [{(x^{9} + x)}^{\frac{1}{6}}]}{{(x^{9} + x)}^{\frac{1}{2 \cdot 3} \cdot 2}}$

Step 2.3.1.2.3

Rewrite the expression.

$\frac{5}{6} \cdot \frac{{(x^{9} + x)}^{\frac{1}{6}} \frac{d}{d x} [9 x^{8} + 1] - (9 x^{8} + 1) \frac{d}{d x} [{(x^{9} + x)}^{\frac{1}{6}}]}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.3.2

By the Sum Rule, the derivative of $9 x^{8} + 1$ with respect to $x$ is $\frac{d}{d x} [9 x^{8}] + \frac{d}{d x} [1]$ .

$\frac{5}{6} \cdot \frac{{(x^{9} + x)}^{\frac{1}{6}} (\frac{d}{d x} [9 x^{8}] + \frac{d}{d x} [1]) - (9 x^{8} + 1) \frac{d}{d x} [{(x^{9} + x)}^{\frac{1}{6}}]}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.3.3

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{8}$ with respect to $x$ is $9 \frac{d}{d x} [x^{8}]$ .

$\frac{5}{6} \cdot \frac{{(x^{9} + x)}^{\frac{1}{6}} (9 \frac{d}{d x} [x^{8}] + \frac{d}{d x} [1]) - (9 x^{8} + 1) \frac{d}{d x} [{(x^{9} + x)}^{\frac{1}{6}}]}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 8$ .

$\frac{5}{6} \cdot \frac{{(x^{9} + x)}^{\frac{1}{6}} (9 (8 x^{7}) + \frac{d}{d x} [1]) - (9 x^{8} + 1) \frac{d}{d x} [{(x^{9} + x)}^{\frac{1}{6}}]}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.3.5

Multiply $8$ by $9$ .

$\frac{5}{6} \cdot \frac{{(x^{9} + x)}^{\frac{1}{6}} (72 x^{7} + \frac{d}{d x} [1]) - (9 x^{8} + 1) \frac{d}{d x} [{(x^{9} + x)}^{\frac{1}{6}}]}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.3.6

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

$\frac{5}{6} \cdot \frac{{(x^{9} + x)}^{\frac{1}{6}} (72 x^{7} + 0) - (9 x^{8} + 1) \frac{d}{d x} [{(x^{9} + x)}^{\frac{1}{6}}]}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.3.7

Simplify the expression.

Tap for more steps...

Step 2.3.7.1

Add $72 x^{7}$ and $0$ .

$\frac{5}{6} \cdot \frac{{(x^{9} + x)}^{\frac{1}{6}} (72 x^{7}) - (9 x^{8} + 1) \frac{d}{d x} [{(x^{9} + x)}^{\frac{1}{6}}]}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.3.7.2

Move $72$ to the left of ${(x^{9} + x)}^{\frac{1}{6}}$ .

$\frac{5}{6} \cdot \frac{72 \cdot {(x^{9} + x)}^{\frac{1}{6}} x^{7} - (9 x^{8} + 1) \frac{d}{d x} [{(x^{9} + x)}^{\frac{1}{6}}]}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{6}}$ and $g (x) = x^{9} + x$ .

Tap for more steps...

Step 2.4.1

To apply the Chain Rule, set $u$ as $x^{9} + x$ .

$\frac{5}{6} \cdot \frac{72 {(x^{9} + x)}^{\frac{1}{6}} x^{7} - (9 x^{8} + 1) (\frac{d}{d u} [u^{\frac{1}{6}}] \frac{d}{d x} [x^{9} + x])}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.4.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{6}$ .

$\frac{5}{6} \cdot \frac{72 {(x^{9} + x)}^{\frac{1}{6}} x^{7} - (9 x^{8} + 1) (\frac{1}{6} u^{\frac{1}{6} - 1} \frac{d}{d x} [x^{9} + x])}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.4.3

Replace all occurrences of $u$ with $x^{9} + x$ .

$\frac{5}{6} \cdot \frac{72 {(x^{9} + x)}^{\frac{1}{6}} x^{7} - (9 x^{8} + 1) (\frac{1}{6} {(x^{9} + x)}^{\frac{1}{6} - 1} \frac{d}{d x} [x^{9} + x])}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{5}{6} \cdot \frac{72 {(x^{9} + x)}^{\frac{1}{6}} x^{7} - (9 x^{8} + 1) (\frac{1}{6} {(x^{9} + x)}^{\frac{1}{6} - 1 \cdot \frac{6}{6}} \frac{d}{d x} [x^{9} + x])}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.6

Combine $- 1$ and $\frac{6}{6}$ .

$\frac{5}{6} \cdot \frac{72 {(x^{9} + x)}^{\frac{1}{6}} x^{7} - (9 x^{8} + 1) (\frac{1}{6} {(x^{9} + x)}^{\frac{1}{6} + \frac{- 1 \cdot 6}{6}} \frac{d}{d x} [x^{9} + x])}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.7

Combine the numerators over the common denominator.

$\frac{5}{6} \cdot \frac{72 {(x^{9} + x)}^{\frac{1}{6}} x^{7} - (9 x^{8} + 1) (\frac{1}{6} {(x^{9} + x)}^{\frac{1 - 1 \cdot 6}{6}} \frac{d}{d x} [x^{9} + x])}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.8

Simplify the numerator.

Tap for more steps...

Step 2.8.1

Multiply $- 1$ by $6$ .

$\frac{5}{6} \cdot \frac{72 {(x^{9} + x)}^{\frac{1}{6}} x^{7} - (9 x^{8} + 1) (\frac{1}{6} {(x^{9} + x)}^{\frac{1 - 6}{6}} \frac{d}{d x} [x^{9} + x])}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.8.2

Subtract $6$ from $1$ .

$\frac{5}{6} \cdot \frac{72 {(x^{9} + x)}^{\frac{1}{6}} x^{7} - (9 x^{8} + 1) (\frac{1}{6} {(x^{9} + x)}^{\frac{- 5}{6}} \frac{d}{d x} [x^{9} + x])}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.9

Combine fractions.

Tap for more steps...

Step 2.9.1

Move the negative in front of the fraction.

$\frac{5}{6} \cdot \frac{72 {(x^{9} + x)}^{\frac{1}{6}} x^{7} - (9 x^{8} + 1) (\frac{1}{6} {(x^{9} + x)}^{- \frac{5}{6}} \frac{d}{d x} [x^{9} + x])}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.9.2

Combine $\frac{1}{6}$ and ${(x^{9} + x)}^{- \frac{5}{6}}$ .

$\frac{5}{6} \cdot \frac{72 {(x^{9} + x)}^{\frac{1}{6}} x^{7} - (9 x^{8} + 1) (\frac{{(x^{9} + x)}^{- \frac{5}{6}}}{6} \frac{d}{d x} [x^{9} + x])}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.9.3

Move ${(x^{9} + x)}^{- \frac{5}{6}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{5}{6} \cdot \frac{72 {(x^{9} + x)}^{\frac{1}{6}} x^{7} - (9 x^{8} + 1) (\frac{1}{6 {(x^{9} + x)}^{\frac{5}{6}}} \frac{d}{d x} [x^{9} + x])}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.10

By the Sum Rule, the derivative of $x^{9} + x$ with respect to $x$ is $\frac{d}{d x} [x^{9}] + \frac{d}{d x} [x]$ .

$\frac{5}{6} \cdot \frac{72 {(x^{9} + x)}^{\frac{1}{6}} x^{7} - (9 x^{8} + 1) (\frac{1}{6 {(x^{9} + x)}^{\frac{5}{6}}} (\frac{d}{d x} [x^{9}] + \frac{d}{d x} [x]))}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.11

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 9$ .

$\frac{5}{6} \cdot \frac{72 {(x^{9} + x)}^{\frac{1}{6}} x^{7} - (9 x^{8} + 1) (\frac{1}{6 {(x^{9} + x)}^{\frac{5}{6}}} (9 x^{8} + \frac{d}{d x} [x]))}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.12

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

$\frac{5}{6} \cdot \frac{72 {(x^{9} + x)}^{\frac{1}{6}} x^{7} - (9 x^{8} + 1) (\frac{1}{6 {(x^{9} + x)}^{\frac{5}{6}}} (9 x^{8} + 1))}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.13

Raise $9 x^{8} + 1$ to the power of $1$ .

$\frac{5}{6} \cdot \frac{72 {(x^{9} + x)}^{\frac{1}{6}} x^{7} - ({(9 x^{8} + 1)}^{1} (9 x^{8} + 1)) \frac{1}{6 {(x^{9} + x)}^{\frac{5}{6}}}}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.14

Raise $9 x^{8} + 1$ to the power of $1$ .

$\frac{5}{6} \cdot \frac{72 {(x^{9} + x)}^{\frac{1}{6}} x^{7} - ({(9 x^{8} + 1)}^{1} {(9 x^{8} + 1)}^{1}) \frac{1}{6 {(x^{9} + x)}^{\frac{5}{6}}}}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.15

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{5}{6} \cdot \frac{72 {(x^{9} + x)}^{\frac{1}{6}} x^{7} - {(9 x^{8} + 1)}^{1 + 1} \frac{1}{6 {(x^{9} + x)}^{\frac{5}{6}}}}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.16

Add $1$ and $1$ .

$\frac{5}{6} \cdot \frac{72 {(x^{9} + x)}^{\frac{1}{6}} x^{7} - {(9 x^{8} + 1)}^{2} \frac{1}{6 {(x^{9} + x)}^{\frac{5}{6}}}}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.17

Combine $\frac{1}{6 {(x^{9} + x)}^{\frac{5}{6}}}$ and ${(9 x^{8} + 1)}^{2}$ .

$\frac{5}{6} \cdot \frac{72 {(x^{9} + x)}^{\frac{1}{6}} x^{7} - \frac{{(9 x^{8} + 1)}^{2}}{6 {(x^{9} + x)}^{\frac{5}{6}}}}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.18

To write $72 {(x^{9} + x)}^{\frac{1}{6}} x^{7}$ as a fraction with a common denominator, multiply by $\frac{6 {(x^{9} + x)}^{\frac{5}{6}}}{6 {(x^{9} + x)}^{\frac{5}{6}}}$ .

$\frac{5}{6} \cdot \frac{72 {(x^{9} + x)}^{\frac{1}{6}} x^{7} \cdot \frac{6 {(x^{9} + x)}^{\frac{5}{6}}}{6 {(x^{9} + x)}^{\frac{5}{6}}} - \frac{{(9 x^{8} + 1)}^{2}}{6 {(x^{9} + x)}^{\frac{5}{6}}}}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.19

Combine $72 {(x^{9} + x)}^{\frac{1}{6}} x^{7}$ and $\frac{6 {(x^{9} + x)}^{\frac{5}{6}}}{6 {(x^{9} + x)}^{\frac{5}{6}}}$ .

$\frac{5}{6} \cdot \frac{\frac{72 {(x^{9} + x)}^{\frac{1}{6}} x^{7} (6 {(x^{9} + x)}^{\frac{5}{6}})}{6 {(x^{9} + x)}^{\frac{5}{6}}} - \frac{{(9 x^{8} + 1)}^{2}}{6 {(x^{9} + x)}^{\frac{5}{6}}}}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.20

Combine the numerators over the common denominator.

$\frac{5}{6} \cdot \frac{\frac{72 {(x^{9} + x)}^{\frac{1}{6}} x^{7} (6 {(x^{9} + x)}^{\frac{5}{6}}) - {(9 x^{8} + 1)}^{2}}{6 {(x^{9} + x)}^{\frac{5}{6}}}}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.21

Multiply $6$ by $72$ .

$\frac{5}{6} \cdot \frac{\frac{432 {(x^{9} + x)}^{\frac{1}{6}} x^{7} {(x^{9} + x)}^{\frac{5}{6}} - {(9 x^{8} + 1)}^{2}}{6 {(x^{9} + x)}^{\frac{5}{6}}}}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.22

Multiply ${(x^{9} + x)}^{\frac{1}{6}}$ by ${(x^{9} + x)}^{\frac{5}{6}}$ by adding the exponents.

Tap for more steps...

Step 2.22.1

Move ${(x^{9} + x)}^{\frac{5}{6}}$ .

$\frac{5}{6} \cdot \frac{\frac{432 ({(x^{9} + x)}^{\frac{5}{6}} {(x^{9} + x)}^{\frac{1}{6}}) x^{7} - {(9 x^{8} + 1)}^{2}}{6 {(x^{9} + x)}^{\frac{5}{6}}}}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.22.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{5}{6} \cdot \frac{\frac{432 {(x^{9} + x)}^{\frac{5}{6} + \frac{1}{6}} x^{7} - {(9 x^{8} + 1)}^{2}}{6 {(x^{9} + x)}^{\frac{5}{6}}}}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.22.3

Combine the numerators over the common denominator.

$\frac{5}{6} \cdot \frac{\frac{432 {(x^{9} + x)}^{\frac{5 + 1}{6}} x^{7} - {(9 x^{8} + 1)}^{2}}{6 {(x^{9} + x)}^{\frac{5}{6}}}}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.22.4

Add $5$ and $1$ .

$\frac{5}{6} \cdot \frac{\frac{432 {(x^{9} + x)}^{\frac{6}{6}} x^{7} - {(9 x^{8} + 1)}^{2}}{6 {(x^{9} + x)}^{\frac{5}{6}}}}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.22.5

Divide $6$ by $6$ .

$\frac{5}{6} \cdot \frac{\frac{432 {(x^{9} + x)}^{1} x^{7} - {(9 x^{8} + 1)}^{2}}{6 {(x^{9} + x)}^{\frac{5}{6}}}}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.23

Simplify $432 {(x^{9} + x)}^{1} x^{7}$ .

$\frac{5}{6} \cdot \frac{\frac{432 (x^{9} + x) x^{7} - {(9 x^{8} + 1)}^{2}}{6 {(x^{9} + x)}^{\frac{5}{6}}}}{{(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.24

Rewrite $\frac{\frac{432 (x^{9} + x) x^{7} - {(9 x^{8} + 1)}^{2}}{6 {(x^{9} + x)}^{\frac{5}{6}}}}{{(x^{9} + x)}^{\frac{1}{3}}}$ as a product.

$\frac{5}{6} (\frac{432 (x^{9} + x) x^{7} - {(9 x^{8} + 1)}^{2}}{6 {(x^{9} + x)}^{\frac{5}{6}}} \cdot \frac{1}{{(x^{9} + x)}^{\frac{1}{3}}})$

Step 2.25

Multiply $\frac{432 (x^{9} + x) x^{7} - {(9 x^{8} + 1)}^{2}}{6 {(x^{9} + x)}^{\frac{5}{6}}}$ by $\frac{1}{{(x^{9} + x)}^{\frac{1}{3}}}$ .

$\frac{5}{6} \cdot \frac{432 (x^{9} + x) x^{7} - {(9 x^{8} + 1)}^{2}}{6 {(x^{9} + x)}^{\frac{5}{6}} {(x^{9} + x)}^{\frac{1}{3}}}$

Step 2.26

Multiply ${(x^{9} + x)}^{\frac{5}{6}}$ by ${(x^{9} + x)}^{\frac{1}{3}}$ by adding the exponents.

Tap for more steps...

Step 2.26.1

Move ${(x^{9} + x)}^{\frac{1}{3}}$ .

$\frac{5}{6} \cdot \frac{432 (x^{9} + x) x^{7} - {(9 x^{8} + 1)}^{2}}{6 ({(x^{9} + x)}^{\frac{1}{3}} {(x^{9} + x)}^{\frac{5}{6}})}$

Step 2.26.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{5}{6} \cdot \frac{432 (x^{9} + x) x^{7} - {(9 x^{8} + 1)}^{2}}{6 {(x^{9} + x)}^{\frac{1}{3} + \frac{5}{6}}}$

Step 2.26.3

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{5}{6} \cdot \frac{432 (x^{9} + x) x^{7} - {(9 x^{8} + 1)}^{2}}{6 {(x^{9} + x)}^{\frac{1}{3} \cdot \frac{2}{2} + \frac{5}{6}}}$

Step 2.26.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 2.26.4.1

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

$\frac{5}{6} \cdot \frac{432 (x^{9} + x) x^{7} - {(9 x^{8} + 1)}^{2}}{6 {(x^{9} + x)}^{\frac{2}{3 \cdot 2} + \frac{5}{6}}}$

Step 2.26.4.2

Multiply $3$ by $2$ .

$\frac{5}{6} \cdot \frac{432 (x^{9} + x) x^{7} - {(9 x^{8} + 1)}^{2}}{6 {(x^{9} + x)}^{\frac{2}{6} + \frac{5}{6}}}$

Step 2.26.5

Combine the numerators over the common denominator.

$\frac{5}{6} \cdot \frac{432 (x^{9} + x) x^{7} - {(9 x^{8} + 1)}^{2}}{6 {(x^{9} + x)}^{\frac{2 + 5}{6}}}$

Step 2.26.6

Add $2$ and $5$ .

$\frac{5}{6} \cdot \frac{432 (x^{9} + x) x^{7} - {(9 x^{8} + 1)}^{2}}{6 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.27

Multiply $\frac{5}{6}$ by $\frac{432 (x^{9} + x) x^{7} - {(9 x^{8} + 1)}^{2}}{6 {(x^{9} + x)}^{\frac{7}{6}}}$ .

$\frac{5 (432 (x^{9} + x) x^{7} - {(9 x^{8} + 1)}^{2})}{6 (6 {(x^{9} + x)}^{\frac{7}{6}})}$

Step 2.28

Multiply $6$ by $6$ .

$\frac{5 (432 (x^{9} + x) x^{7} - {(9 x^{8} + 1)}^{2})}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29

Simplify.

Tap for more steps...

Step 2.29.1

Apply the distributive property.

$\frac{5 ((432 x^{9} + 432 x) x^{7} - {(9 x^{8} + 1)}^{2})}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.2

Apply the distributive property.

$\frac{5 (432 x^{9} x^{7} + 432 x \cdot x^{7} - {(9 x^{8} + 1)}^{2})}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.3

Apply the distributive property.

$\frac{5 (432 x^{9} x^{7}) + 5 (432 x \cdot x^{7}) + 5 (- {(9 x^{8} + 1)}^{2})}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4

Simplify the numerator.

Tap for more steps...

Step 2.29.4.1

Simplify each term.

Tap for more steps...

Step 2.29.4.1.1

Multiply $x^{9}$ by $x^{7}$ by adding the exponents.

Tap for more steps...

Step 2.29.4.1.1.1

Move $x^{7}$ .

$\frac{5 (432 (x^{7} x^{9})) + 5 (432 x \cdot x^{7}) + 5 (- {(9 x^{8} + 1)}^{2})}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{5 (432 x^{7 + 9}) + 5 (432 x \cdot x^{7}) + 5 (- {(9 x^{8} + 1)}^{2})}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.1.3

Add $7$ and $9$ .

$\frac{5 (432 x^{16}) + 5 (432 x \cdot x^{7}) + 5 (- {(9 x^{8} + 1)}^{2})}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.2

Multiply $432$ by $5$ .

$\frac{2160 x^{16} + 5 (432 x \cdot x^{7}) + 5 (- {(9 x^{8} + 1)}^{2})}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.3

Multiply $x$ by $x^{7}$ by adding the exponents.

Tap for more steps...

Step 2.29.4.1.3.1

Move $x^{7}$ .

$\frac{2160 x^{16} + 5 (432 (x^{7} x)) + 5 (- {(9 x^{8} + 1)}^{2})}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.3.2

Multiply $x^{7}$ by $x$ .

Tap for more steps...

Step 2.29.4.1.3.2.1

Raise $x$ to the power of $1$ .

$\frac{2160 x^{16} + 5 (432 (x^{7} x^{1})) + 5 (- {(9 x^{8} + 1)}^{2})}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2160 x^{16} + 5 (432 x^{7 + 1}) + 5 (- {(9 x^{8} + 1)}^{2})}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.3.3

Add $7$ and $1$ .

$\frac{2160 x^{16} + 5 (432 x^{8}) + 5 (- {(9 x^{8} + 1)}^{2})}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.4

Multiply $432$ by $5$ .

$\frac{2160 x^{16} + 2160 x^{8} + 5 (- {(9 x^{8} + 1)}^{2})}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.5

Rewrite ${(9 x^{8} + 1)}^{2}$ as $(9 x^{8} + 1) (9 x^{8} + 1)$ .

$\frac{2160 x^{16} + 2160 x^{8} + 5 (- ((9 x^{8} + 1) (9 x^{8} + 1)))}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.6

Expand $(9 x^{8} + 1) (9 x^{8} + 1)$ using the FOIL Method.

Tap for more steps...

Step 2.29.4.1.6.1

Apply the distributive property.

$\frac{2160 x^{16} + 2160 x^{8} + 5 (- (9 x^{8} (9 x^{8} + 1) + 1 (9 x^{8} + 1)))}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.6.2

Apply the distributive property.

$\frac{2160 x^{16} + 2160 x^{8} + 5 (- (9 x^{8} (9 x^{8}) + 9 x^{8} \cdot 1 + 1 (9 x^{8} + 1)))}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.6.3

Apply the distributive property.

$\frac{2160 x^{16} + 2160 x^{8} + 5 (- (9 x^{8} (9 x^{8}) + 9 x^{8} \cdot 1 + 1 (9 x^{8}) + 1 \cdot 1))}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.7

Simplify and combine like terms.

Tap for more steps...

Step 2.29.4.1.7.1

Simplify each term.

Tap for more steps...

Step 2.29.4.1.7.1.1

Rewrite using the commutative property of multiplication.

$\frac{2160 x^{16} + 2160 x^{8} + 5 (- (9 \cdot 9 x^{8} x^{8} + 9 x^{8} \cdot 1 + 1 (9 x^{8}) + 1 \cdot 1))}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.7.1.2

Multiply $x^{8}$ by $x^{8}$ by adding the exponents.

Tap for more steps...

Step 2.29.4.1.7.1.2.1

Move $x^{8}$ .

$\frac{2160 x^{16} + 2160 x^{8} + 5 (- (9 \cdot 9 (x^{8} x^{8}) + 9 x^{8} \cdot 1 + 1 (9 x^{8}) + 1 \cdot 1))}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.7.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2160 x^{16} + 2160 x^{8} + 5 (- (9 \cdot 9 x^{8 + 8} + 9 x^{8} \cdot 1 + 1 (9 x^{8}) + 1 \cdot 1))}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.7.1.2.3

Add $8$ and $8$ .

$\frac{2160 x^{16} + 2160 x^{8} + 5 (- (9 \cdot 9 x^{16} + 9 x^{8} \cdot 1 + 1 (9 x^{8}) + 1 \cdot 1))}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.7.1.3

Multiply $9$ by $9$ .

$\frac{2160 x^{16} + 2160 x^{8} + 5 (- (81 x^{16} + 9 x^{8} \cdot 1 + 1 (9 x^{8}) + 1 \cdot 1))}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.7.1.4

Multiply $9$ by $1$ .

$\frac{2160 x^{16} + 2160 x^{8} + 5 (- (81 x^{16} + 9 x^{8} + 1 (9 x^{8}) + 1 \cdot 1))}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.7.1.5

Multiply $9 x^{8}$ by $1$ .

$\frac{2160 x^{16} + 2160 x^{8} + 5 (- (81 x^{16} + 9 x^{8} + 9 x^{8} + 1 \cdot 1))}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.7.1.6

Multiply $1$ by $1$ .

$\frac{2160 x^{16} + 2160 x^{8} + 5 (- (81 x^{16} + 9 x^{8} + 9 x^{8} + 1))}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.7.2

Add $9 x^{8}$ and $9 x^{8}$ .

$\frac{2160 x^{16} + 2160 x^{8} + 5 (- (81 x^{16} + 18 x^{8} + 1))}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.8

Apply the distributive property.

$\frac{2160 x^{16} + 2160 x^{8} + 5 (- (81 x^{16}) - (18 x^{8}) - 1 \cdot 1)}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.9

Simplify.

Tap for more steps...

Step 2.29.4.1.9.1

Multiply $81$ by $- 1$ .

$\frac{2160 x^{16} + 2160 x^{8} + 5 (- 81 x^{16} - (18 x^{8}) - 1 \cdot 1)}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.9.2

Multiply $18$ by $- 1$ .

$\frac{2160 x^{16} + 2160 x^{8} + 5 (- 81 x^{16} - 18 x^{8} - 1 \cdot 1)}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.9.3

Multiply $- 1$ by $1$ .

$\frac{2160 x^{16} + 2160 x^{8} + 5 (- 81 x^{16} - 18 x^{8} - 1)}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.10

Apply the distributive property.

$\frac{2160 x^{16} + 2160 x^{8} + 5 (- 81 x^{16}) + 5 (- 18 x^{8}) + 5 \cdot - 1}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.11

Simplify.

Tap for more steps...

Step 2.29.4.1.11.1

Multiply $- 81$ by $5$ .

$\frac{2160 x^{16} + 2160 x^{8} - 405 x^{16} + 5 (- 18 x^{8}) + 5 \cdot - 1}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.11.2

Multiply $- 18$ by $5$ .

$\frac{2160 x^{16} + 2160 x^{8} - 405 x^{16} - 90 x^{8} + 5 \cdot - 1}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.1.11.3

Multiply $5$ by $- 1$ .

$\frac{2160 x^{16} + 2160 x^{8} - 405 x^{16} - 90 x^{8} - 5}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.2

Subtract $405 x^{16}$ from $2160 x^{16}$ .

$\frac{1755 x^{16} + 2160 x^{8} - 90 x^{8} - 5}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.4.3

Subtract $90 x^{8}$ from $2160 x^{8}$ .

$\frac{1755 x^{16} + 2070 x^{8} - 5}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.5

Factor $5$ out of $1755 x^{16} + 2070 x^{8} - 5$ .

Tap for more steps...

Step 2.29.5.1

Factor $5$ out of $1755 x^{16}$ .

$\frac{5 (351 x^{16}) + 2070 x^{8} - 5}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.5.2

Factor $5$ out of $2070 x^{8}$ .

$\frac{5 (351 x^{16}) + 5 (414 x^{8}) - 5}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.5.3

Factor $5$ out of $- 5$ .

$\frac{5 (351 x^{16}) + 5 (414 x^{8}) + 5 (- 1)}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.5.4

Factor $5$ out of $5 (351 x^{16}) + 5 (414 x^{8})$ .

$\frac{5 (351 x^{16} + 414 x^{8}) + 5 (- 1)}{36 {(x^{9} + x)}^{\frac{7}{6}}}$

Step 2.29.5.5

Factor $5$ out of $5 (351 x^{16} + 414 x^{8}) + 5 (- 1)$ .

$f'' (x) = \frac{5 (351 x^{16} + 414 x^{8} - 1)}{36 {(x^{9} + x)}^{\frac{7}{6}}}$