Calculus Examples

$y = {(x^{6} - x)}^{\frac{3}{4}}$

Step 1

Find the first derivative.

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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 $f (x) = x^{\frac{3}{4}}$ and $g (x) = x^{6} - x$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $x^{6} - x$ .

$\frac{d}{d u} [u^{\frac{3}{4}}] \frac{d}{d x} [x^{6} - x]$

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{3}{4}$ .

$\frac{3}{4} u^{\frac{3}{4} - 1} \frac{d}{d x} [x^{6} - x]$

Step 1.1.3

Replace all occurrences of $u$ with $x^{6} - x$ .

$\frac{3}{4} {(x^{6} - x)}^{\frac{3}{4} - 1} \frac{d}{d x} [x^{6} - x]$

Step 1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{3}{4} {(x^{6} - x)}^{\frac{3}{4} - 1 \cdot \frac{4}{4}} \frac{d}{d x} [x^{6} - x]$

Step 1.3

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

$\frac{3}{4} {(x^{6} - x)}^{\frac{3}{4} + \frac{- 1 \cdot 4}{4}} \frac{d}{d x} [x^{6} - x]$

Step 1.4

Combine the numerators over the common denominator.

$\frac{3}{4} {(x^{6} - x)}^{\frac{3 - 1 \cdot 4}{4}} \frac{d}{d x} [x^{6} - x]$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $- 1$ by $4$ .

$\frac{3}{4} {(x^{6} - x)}^{\frac{3 - 4}{4}} \frac{d}{d x} [x^{6} - x]$

Step 1.5.2

Subtract $4$ from $3$ .

$\frac{3}{4} {(x^{6} - x)}^{\frac{- 1}{4}} \frac{d}{d x} [x^{6} - x]$

Step 1.6

Combine fractions.

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Step 1.6.1

Move the negative in front of the fraction.

$\frac{3}{4} {(x^{6} - x)}^{- \frac{1}{4}} \frac{d}{d x} [x^{6} - x]$

Step 1.6.2

Combine $\frac{3}{4}$ and ${(x^{6} - x)}^{- \frac{1}{4}}$ .

$\frac{3 {(x^{6} - x)}^{- \frac{1}{4}}}{4} \frac{d}{d x} [x^{6} - x]$

Step 1.6.3

Move ${(x^{6} - x)}^{- \frac{1}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{3}{4 {(x^{6} - x)}^{\frac{1}{4}}} \frac{d}{d x} [x^{6} - x]$

Step 1.7

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

$\frac{3}{4 {(x^{6} - x)}^{\frac{1}{4}}} (\frac{d}{d x} [x^{6}] + \frac{d}{d x} [- x])$

Step 1.8

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

$\frac{3}{4 {(x^{6} - x)}^{\frac{1}{4}}} (6 x^{5} + \frac{d}{d x} [- x])$

Step 1.9

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

$\frac{3}{4 {(x^{6} - x)}^{\frac{1}{4}}} (6 x^{5} - \frac{d}{d x} [x])$

Step 1.10

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

$\frac{3}{4 {(x^{6} - x)}^{\frac{1}{4}}} (6 x^{5} - 1 \cdot 1)$

Step 1.11

Multiply $- 1$ by $1$ .

$\frac{3}{4 {(x^{6} - x)}^{\frac{1}{4}}} (6 x^{5} - 1)$

Step 1.12

Simplify.

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Step 1.12.1

Reorder the factors of $\frac{3}{4 {(x^{6} - x)}^{\frac{1}{4}}} (6 x^{5} - 1)$ .

$(6 x^{5} - 1) \frac{3}{4 {(x^{6} - x)}^{\frac{1}{4}}}$

Step 1.12.2

Multiply $6 x^{5} - 1$ by $\frac{3}{4 {(x^{6} - x)}^{\frac{1}{4}}}$ .

$\frac{(6 x^{5} - 1) \cdot 3}{4 {(x^{6} - x)}^{\frac{1}{4}}}$

Step 1.12.3

Move $3$ to the left of $6 x^{5} - 1$ .

$f' (x) = \frac{3 (6 x^{5} - 1)}{4 {(x^{6} - x)}^{\frac{1}{4}}}$

Step 2

Find the second derivative.

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Step 2.1

Since $\frac{3}{4}$ is constant with respect to $x$ , the derivative of $\frac{3 (6 x^{5} - 1)}{4 {(x^{6} - x)}^{\frac{1}{4}}}$ with respect to $x$ is $\frac{3}{4} \frac{d}{d x} [\frac{6 x^{5} - 1}{{(x^{6} - x)}^{\frac{1}{4}}}]$ .

$\frac{3}{4} \frac{d}{d x} [\frac{6 x^{5} - 1}{{(x^{6} - x)}^{\frac{1}{4}}}]$

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 $f (x) = 6 x^{5} - 1$ and $g (x) = {(x^{6} - x)}^{\frac{1}{4}}$ .

$\frac{3}{4} \cdot \frac{{(x^{6} - x)}^{\frac{1}{4}} \frac{d}{d x} [6 x^{5} - 1] - (6 x^{5} - 1) \frac{d}{d x} [{(x^{6} - x)}^{\frac{1}{4}}]}{{({(x^{6} - x)}^{\frac{1}{4}})}^{2}}$

Step 2.3

Differentiate.

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Step 2.3.1

Multiply the exponents in ${({(x^{6} - x)}^{\frac{1}{4}})}^{2}$ .

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Step 2.3.1.1

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

$\frac{3}{4} \cdot \frac{{(x^{6} - x)}^{\frac{1}{4}} \frac{d}{d x} [6 x^{5} - 1] - (6 x^{5} - 1) \frac{d}{d x} [{(x^{6} - x)}^{\frac{1}{4}}]}{{(x^{6} - x)}^{\frac{1}{4} \cdot 2}}$

Step 2.3.1.2

Cancel the common factor of $2$ .

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Step 2.3.1.2.1

Factor $2$ out of $4$ .

$\frac{3}{4} \cdot \frac{{(x^{6} - x)}^{\frac{1}{4}} \frac{d}{d x} [6 x^{5} - 1] - (6 x^{5} - 1) \frac{d}{d x} [{(x^{6} - x)}^{\frac{1}{4}}]}{{(x^{6} - x)}^{\frac{1}{2 (2)} \cdot 2}}$

Step 2.3.1.2.2

Cancel the common factor.

$\frac{3}{4} \cdot \frac{{(x^{6} - x)}^{\frac{1}{4}} \frac{d}{d x} [6 x^{5} - 1] - (6 x^{5} - 1) \frac{d}{d x} [{(x^{6} - x)}^{\frac{1}{4}}]}{{(x^{6} - x)}^{\frac{1}{2 \cdot 2} \cdot 2}}$

Step 2.3.1.2.3

Rewrite the expression.

$\frac{3}{4} \cdot \frac{{(x^{6} - x)}^{\frac{1}{4}} \frac{d}{d x} [6 x^{5} - 1] - (6 x^{5} - 1) \frac{d}{d x} [{(x^{6} - x)}^{\frac{1}{4}}]}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.3.2

By the Sum Rule, the derivative of $6 x^{5} - 1$ with respect to $x$ is $\frac{d}{d x} [6 x^{5}] + \frac{d}{d x} [- 1]$ .

$\frac{3}{4} \cdot \frac{{(x^{6} - x)}^{\frac{1}{4}} (\frac{d}{d x} [6 x^{5}] + \frac{d}{d x} [- 1]) - (6 x^{5} - 1) \frac{d}{d x} [{(x^{6} - x)}^{\frac{1}{4}}]}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.3.3

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

$\frac{3}{4} \cdot \frac{{(x^{6} - x)}^{\frac{1}{4}} (6 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 1]) - (6 x^{5} - 1) \frac{d}{d x} [{(x^{6} - x)}^{\frac{1}{4}}]}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.3.4

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

$\frac{3}{4} \cdot \frac{{(x^{6} - x)}^{\frac{1}{4}} (6 (5 x^{4}) + \frac{d}{d x} [- 1]) - (6 x^{5} - 1) \frac{d}{d x} [{(x^{6} - x)}^{\frac{1}{4}}]}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.3.5

Multiply $5$ by $6$ .

$\frac{3}{4} \cdot \frac{{(x^{6} - x)}^{\frac{1}{4}} (30 x^{4} + \frac{d}{d x} [- 1]) - (6 x^{5} - 1) \frac{d}{d x} [{(x^{6} - x)}^{\frac{1}{4}}]}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.3.6

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

$\frac{3}{4} \cdot \frac{{(x^{6} - x)}^{\frac{1}{4}} (30 x^{4} + 0) - (6 x^{5} - 1) \frac{d}{d x} [{(x^{6} - x)}^{\frac{1}{4}}]}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.3.7

Simplify the expression.

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Step 2.3.7.1

Add $30 x^{4}$ and $0$ .

$\frac{3}{4} \cdot \frac{{(x^{6} - x)}^{\frac{1}{4}} (30 x^{4}) - (6 x^{5} - 1) \frac{d}{d x} [{(x^{6} - x)}^{\frac{1}{4}}]}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.3.7.2

Move $30$ to the left of ${(x^{6} - x)}^{\frac{1}{4}}$ .

$\frac{3}{4} \cdot \frac{30 \cdot {(x^{6} - x)}^{\frac{1}{4}} x^{4} - (6 x^{5} - 1) \frac{d}{d x} [{(x^{6} - x)}^{\frac{1}{4}}]}{{(x^{6} - x)}^{\frac{1}{2}}}$

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 $f (x) = x^{\frac{1}{4}}$ and $g (x) = x^{6} - x$ .

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Step 2.4.1

To apply the Chain Rule, set $u$ as $x^{6} - x$ .

$\frac{3}{4} \cdot \frac{30 {(x^{6} - x)}^{\frac{1}{4}} x^{4} - (6 x^{5} - 1) (\frac{d}{d u} [u^{\frac{1}{4}}] \frac{d}{d x} [x^{6} - x])}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.4.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{4}$ .

$\frac{3}{4} \cdot \frac{30 {(x^{6} - x)}^{\frac{1}{4}} x^{4} - (6 x^{5} - 1) (\frac{1}{4} u^{\frac{1}{4} - 1} \frac{d}{d x} [x^{6} - x])}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.4.3

Replace all occurrences of $u$ with $x^{6} - x$ .

$\frac{3}{4} \cdot \frac{30 {(x^{6} - x)}^{\frac{1}{4}} x^{4} - (6 x^{5} - 1) (\frac{1}{4} {(x^{6} - x)}^{\frac{1}{4} - 1} \frac{d}{d x} [x^{6} - x])}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{3}{4} \cdot \frac{30 {(x^{6} - x)}^{\frac{1}{4}} x^{4} - (6 x^{5} - 1) (\frac{1}{4} {(x^{6} - x)}^{\frac{1}{4} - 1 \cdot \frac{4}{4}} \frac{d}{d x} [x^{6} - x])}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.6

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

$\frac{3}{4} \cdot \frac{30 {(x^{6} - x)}^{\frac{1}{4}} x^{4} - (6 x^{5} - 1) (\frac{1}{4} {(x^{6} - x)}^{\frac{1}{4} + \frac{- 1 \cdot 4}{4}} \frac{d}{d x} [x^{6} - x])}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.7

Combine the numerators over the common denominator.

$\frac{3}{4} \cdot \frac{30 {(x^{6} - x)}^{\frac{1}{4}} x^{4} - (6 x^{5} - 1) (\frac{1}{4} {(x^{6} - x)}^{\frac{1 - 1 \cdot 4}{4}} \frac{d}{d x} [x^{6} - x])}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.8

Simplify the numerator.

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Step 2.8.1

Multiply $- 1$ by $4$ .

$\frac{3}{4} \cdot \frac{30 {(x^{6} - x)}^{\frac{1}{4}} x^{4} - (6 x^{5} - 1) (\frac{1}{4} {(x^{6} - x)}^{\frac{1 - 4}{4}} \frac{d}{d x} [x^{6} - x])}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.8.2

Subtract $4$ from $1$ .

$\frac{3}{4} \cdot \frac{30 {(x^{6} - x)}^{\frac{1}{4}} x^{4} - (6 x^{5} - 1) (\frac{1}{4} {(x^{6} - x)}^{\frac{- 3}{4}} \frac{d}{d x} [x^{6} - x])}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.9

Combine fractions.

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Step 2.9.1

Move the negative in front of the fraction.

$\frac{3}{4} \cdot \frac{30 {(x^{6} - x)}^{\frac{1}{4}} x^{4} - (6 x^{5} - 1) (\frac{1}{4} {(x^{6} - x)}^{- \frac{3}{4}} \frac{d}{d x} [x^{6} - x])}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.9.2

Combine $\frac{1}{4}$ and ${(x^{6} - x)}^{- \frac{3}{4}}$ .

$\frac{3}{4} \cdot \frac{30 {(x^{6} - x)}^{\frac{1}{4}} x^{4} - (6 x^{5} - 1) (\frac{{(x^{6} - x)}^{- \frac{3}{4}}}{4} \frac{d}{d x} [x^{6} - x])}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.9.3

Move ${(x^{6} - x)}^{- \frac{3}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{3}{4} \cdot \frac{30 {(x^{6} - x)}^{\frac{1}{4}} x^{4} - (6 x^{5} - 1) (\frac{1}{4 {(x^{6} - x)}^{\frac{3}{4}}} \frac{d}{d x} [x^{6} - x])}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.10

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

$\frac{3}{4} \cdot \frac{30 {(x^{6} - x)}^{\frac{1}{4}} x^{4} - (6 x^{5} - 1) (\frac{1}{4 {(x^{6} - x)}^{\frac{3}{4}}} (\frac{d}{d x} [x^{6}] + \frac{d}{d x} [- x]))}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.11

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

$\frac{3}{4} \cdot \frac{30 {(x^{6} - x)}^{\frac{1}{4}} x^{4} - (6 x^{5} - 1) (\frac{1}{4 {(x^{6} - x)}^{\frac{3}{4}}} (6 x^{5} + \frac{d}{d x} [- x]))}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.12

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

$\frac{3}{4} \cdot \frac{30 {(x^{6} - x)}^{\frac{1}{4}} x^{4} - (6 x^{5} - 1) (\frac{1}{4 {(x^{6} - x)}^{\frac{3}{4}}} (6 x^{5} - \frac{d}{d x} [x]))}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.13

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

$\frac{3}{4} \cdot \frac{30 {(x^{6} - x)}^{\frac{1}{4}} x^{4} - (6 x^{5} - 1) (\frac{1}{4 {(x^{6} - x)}^{\frac{3}{4}}} (6 x^{5} - 1 \cdot 1))}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.14

Multiply $- 1$ by $1$ .

$\frac{3}{4} \cdot \frac{30 {(x^{6} - x)}^{\frac{1}{4}} x^{4} - (6 x^{5} - 1) (\frac{1}{4 {(x^{6} - x)}^{\frac{3}{4}}} (6 x^{5} - 1))}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.15

Raise $6 x^{5} - 1$ to the power of $1$ .

$\frac{3}{4} \cdot \frac{30 {(x^{6} - x)}^{\frac{1}{4}} x^{4} - ({(6 x^{5} - 1)}^{1} (6 x^{5} - 1)) \frac{1}{4 {(x^{6} - x)}^{\frac{3}{4}}}}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.16

Raise $6 x^{5} - 1$ to the power of $1$ .

$\frac{3}{4} \cdot \frac{30 {(x^{6} - x)}^{\frac{1}{4}} x^{4} - ({(6 x^{5} - 1)}^{1} {(6 x^{5} - 1)}^{1}) \frac{1}{4 {(x^{6} - x)}^{\frac{3}{4}}}}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.17

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

$\frac{3}{4} \cdot \frac{30 {(x^{6} - x)}^{\frac{1}{4}} x^{4} - {(6 x^{5} - 1)}^{1 + 1} \frac{1}{4 {(x^{6} - x)}^{\frac{3}{4}}}}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.18

Add $1$ and $1$ .

$\frac{3}{4} \cdot \frac{30 {(x^{6} - x)}^{\frac{1}{4}} x^{4} - {(6 x^{5} - 1)}^{2} \frac{1}{4 {(x^{6} - x)}^{\frac{3}{4}}}}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.19

Combine $\frac{1}{4 {(x^{6} - x)}^{\frac{3}{4}}}$ and ${(6 x^{5} - 1)}^{2}$ .

$\frac{3}{4} \cdot \frac{30 {(x^{6} - x)}^{\frac{1}{4}} x^{4} - \frac{{(6 x^{5} - 1)}^{2}}{4 {(x^{6} - x)}^{\frac{3}{4}}}}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.20

To write $30 {(x^{6} - x)}^{\frac{1}{4}} x^{4}$ as a fraction with a common denominator, multiply by $\frac{4 {(x^{6} - x)}^{\frac{3}{4}}}{4 {(x^{6} - x)}^{\frac{3}{4}}}$ .

$\frac{3}{4} \cdot \frac{30 {(x^{6} - x)}^{\frac{1}{4}} x^{4} \cdot \frac{4 {(x^{6} - x)}^{\frac{3}{4}}}{4 {(x^{6} - x)}^{\frac{3}{4}}} - \frac{{(6 x^{5} - 1)}^{2}}{4 {(x^{6} - x)}^{\frac{3}{4}}}}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.21

Combine $30 {(x^{6} - x)}^{\frac{1}{4}} x^{4}$ and $\frac{4 {(x^{6} - x)}^{\frac{3}{4}}}{4 {(x^{6} - x)}^{\frac{3}{4}}}$ .

$\frac{3}{4} \cdot \frac{\frac{30 {(x^{6} - x)}^{\frac{1}{4}} x^{4} (4 {(x^{6} - x)}^{\frac{3}{4}})}{4 {(x^{6} - x)}^{\frac{3}{4}}} - \frac{{(6 x^{5} - 1)}^{2}}{4 {(x^{6} - x)}^{\frac{3}{4}}}}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.22

Combine the numerators over the common denominator.

$\frac{3}{4} \cdot \frac{\frac{30 {(x^{6} - x)}^{\frac{1}{4}} x^{4} (4 {(x^{6} - x)}^{\frac{3}{4}}) - {(6 x^{5} - 1)}^{2}}{4 {(x^{6} - x)}^{\frac{3}{4}}}}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.23

Multiply $4$ by $30$ .

$\frac{3}{4} \cdot \frac{\frac{120 {(x^{6} - x)}^{\frac{1}{4}} x^{4} {(x^{6} - x)}^{\frac{3}{4}} - {(6 x^{5} - 1)}^{2}}{4 {(x^{6} - x)}^{\frac{3}{4}}}}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.24

Multiply ${(x^{6} - x)}^{\frac{1}{4}}$ by ${(x^{6} - x)}^{\frac{3}{4}}$ by adding the exponents.

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Step 2.24.1

Move ${(x^{6} - x)}^{\frac{3}{4}}$ .

$\frac{3}{4} \cdot \frac{\frac{120 ({(x^{6} - x)}^{\frac{3}{4}} {(x^{6} - x)}^{\frac{1}{4}}) x^{4} - {(6 x^{5} - 1)}^{2}}{4 {(x^{6} - x)}^{\frac{3}{4}}}}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.24.2

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

$\frac{3}{4} \cdot \frac{\frac{120 {(x^{6} - x)}^{\frac{3}{4} + \frac{1}{4}} x^{4} - {(6 x^{5} - 1)}^{2}}{4 {(x^{6} - x)}^{\frac{3}{4}}}}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.24.3

Combine the numerators over the common denominator.

$\frac{3}{4} \cdot \frac{\frac{120 {(x^{6} - x)}^{\frac{3 + 1}{4}} x^{4} - {(6 x^{5} - 1)}^{2}}{4 {(x^{6} - x)}^{\frac{3}{4}}}}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.24.4

Add $3$ and $1$ .

$\frac{3}{4} \cdot \frac{\frac{120 {(x^{6} - x)}^{\frac{4}{4}} x^{4} - {(6 x^{5} - 1)}^{2}}{4 {(x^{6} - x)}^{\frac{3}{4}}}}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.24.5

Divide $4$ by $4$ .

$\frac{3}{4} \cdot \frac{\frac{120 {(x^{6} - x)}^{1} x^{4} - {(6 x^{5} - 1)}^{2}}{4 {(x^{6} - x)}^{\frac{3}{4}}}}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.25

Simplify $120 {(x^{6} - x)}^{1} x^{4}$ .

$\frac{3}{4} \cdot \frac{\frac{120 (x^{6} - x) x^{4} - {(6 x^{5} - 1)}^{2}}{4 {(x^{6} - x)}^{\frac{3}{4}}}}{{(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.26

Rewrite $\frac{\frac{120 (x^{6} - x) x^{4} - {(6 x^{5} - 1)}^{2}}{4 {(x^{6} - x)}^{\frac{3}{4}}}}{{(x^{6} - x)}^{\frac{1}{2}}}$ as a product.

$\frac{3}{4} (\frac{120 (x^{6} - x) x^{4} - {(6 x^{5} - 1)}^{2}}{4 {(x^{6} - x)}^{\frac{3}{4}}} \cdot \frac{1}{{(x^{6} - x)}^{\frac{1}{2}}})$

Step 2.27

Multiply $\frac{120 (x^{6} - x) x^{4} - {(6 x^{5} - 1)}^{2}}{4 {(x^{6} - x)}^{\frac{3}{4}}}$ by $\frac{1}{{(x^{6} - x)}^{\frac{1}{2}}}$ .

$\frac{3}{4} \cdot \frac{120 (x^{6} - x) x^{4} - {(6 x^{5} - 1)}^{2}}{4 {(x^{6} - x)}^{\frac{3}{4}} {(x^{6} - x)}^{\frac{1}{2}}}$

Step 2.28

Multiply ${(x^{6} - x)}^{\frac{3}{4}}$ by ${(x^{6} - x)}^{\frac{1}{2}}$ by adding the exponents.

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Step 2.28.1

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

$\frac{3}{4} \cdot \frac{120 (x^{6} - x) x^{4} - {(6 x^{5} - 1)}^{2}}{4 ({(x^{6} - x)}^{\frac{1}{2}} {(x^{6} - x)}^{\frac{3}{4}})}$

Step 2.28.2

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

$\frac{3}{4} \cdot \frac{120 (x^{6} - x) x^{4} - {(6 x^{5} - 1)}^{2}}{4 {(x^{6} - x)}^{\frac{1}{2} + \frac{3}{4}}}$

Step 2.28.3

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

$\frac{3}{4} \cdot \frac{120 (x^{6} - x) x^{4} - {(6 x^{5} - 1)}^{2}}{4 {(x^{6} - x)}^{\frac{1}{2} \cdot \frac{2}{2} + \frac{3}{4}}}$

Step 2.28.4

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.28.4.1

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

$\frac{3}{4} \cdot \frac{120 (x^{6} - x) x^{4} - {(6 x^{5} - 1)}^{2}}{4 {(x^{6} - x)}^{\frac{2}{2 \cdot 2} + \frac{3}{4}}}$

Step 2.28.4.2

Multiply $2$ by $2$ .

$\frac{3}{4} \cdot \frac{120 (x^{6} - x) x^{4} - {(6 x^{5} - 1)}^{2}}{4 {(x^{6} - x)}^{\frac{2}{4} + \frac{3}{4}}}$

Step 2.28.5

Combine the numerators over the common denominator.

$\frac{3}{4} \cdot \frac{120 (x^{6} - x) x^{4} - {(6 x^{5} - 1)}^{2}}{4 {(x^{6} - x)}^{\frac{2 + 3}{4}}}$

Step 2.28.6

Add $2$ and $3$ .

$\frac{3}{4} \cdot \frac{120 (x^{6} - x) x^{4} - {(6 x^{5} - 1)}^{2}}{4 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.29

Multiply $\frac{3}{4}$ by $\frac{120 (x^{6} - x) x^{4} - {(6 x^{5} - 1)}^{2}}{4 {(x^{6} - x)}^{\frac{5}{4}}}$ .

$\frac{3 (120 (x^{6} - x) x^{4} - {(6 x^{5} - 1)}^{2})}{4 (4 {(x^{6} - x)}^{\frac{5}{4}})}$

Step 2.30

Multiply $4$ by $4$ .

$\frac{3 (120 (x^{6} - x) x^{4} - {(6 x^{5} - 1)}^{2})}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31

Simplify.

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Step 2.31.1

Apply the distributive property.

$\frac{3 ((120 x^{6} + 120 (- x)) x^{4} - {(6 x^{5} - 1)}^{2})}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.2

Apply the distributive property.

$\frac{3 (120 x^{6} x^{4} + 120 (- x) x^{4} - {(6 x^{5} - 1)}^{2})}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.3

Apply the distributive property.

$\frac{3 (120 x^{6} x^{4}) + 3 (120 (- x) x^{4}) + 3 (- {(6 x^{5} - 1)}^{2})}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4

Simplify the numerator.

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Step 2.31.4.1

Simplify each term.

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Step 2.31.4.1.1

Multiply $x^{6}$ by $x^{4}$ by adding the exponents.

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Step 2.31.4.1.1.1

Move $x^{4}$ .

$\frac{3 (120 (x^{4} x^{6})) + 3 (120 (- x) x^{4}) + 3 (- {(6 x^{5} - 1)}^{2})}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.1.2

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

$\frac{3 (120 x^{4 + 6}) + 3 (120 (- x) x^{4}) + 3 (- {(6 x^{5} - 1)}^{2})}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.1.3

Add $4$ and $6$ .

$\frac{3 (120 x^{10}) + 3 (120 (- x) x^{4}) + 3 (- {(6 x^{5} - 1)}^{2})}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.2

Multiply $120$ by $3$ .

$\frac{360 x^{10} + 3 (120 (- x) x^{4}) + 3 (- {(6 x^{5} - 1)}^{2})}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.3

Multiply $x$ by $x^{4}$ by adding the exponents.

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Step 2.31.4.1.3.1

Move $x^{4}$ .

$\frac{360 x^{10} + 3 (120 (- (x^{4} x))) + 3 (- {(6 x^{5} - 1)}^{2})}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.3.2

Multiply $x^{4}$ by $x$ .

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Step 2.31.4.1.3.2.1

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

$\frac{360 x^{10} + 3 (120 (- (x^{4} x^{1}))) + 3 (- {(6 x^{5} - 1)}^{2})}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.3.2.2

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

$\frac{360 x^{10} + 3 (120 (- x^{4 + 1})) + 3 (- {(6 x^{5} - 1)}^{2})}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.3.3

Add $4$ and $1$ .

$\frac{360 x^{10} + 3 (120 (- x^{5})) + 3 (- {(6 x^{5} - 1)}^{2})}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.4

Multiply $- 1$ by $120$ .

$\frac{360 x^{10} + 3 (- 120 x^{5}) + 3 (- {(6 x^{5} - 1)}^{2})}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.5

Multiply $- 120$ by $3$ .

$\frac{360 x^{10} - 360 x^{5} + 3 (- {(6 x^{5} - 1)}^{2})}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.6

Rewrite ${(6 x^{5} - 1)}^{2}$ as $(6 x^{5} - 1) (6 x^{5} - 1)$ .

$\frac{360 x^{10} - 360 x^{5} + 3 (- ((6 x^{5} - 1) (6 x^{5} - 1)))}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.7

Expand $(6 x^{5} - 1) (6 x^{5} - 1)$ using the FOIL Method.

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Step 2.31.4.1.7.1

Apply the distributive property.

$\frac{360 x^{10} - 360 x^{5} + 3 (- (6 x^{5} (6 x^{5} - 1) - 1 (6 x^{5} - 1)))}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.7.2

Apply the distributive property.

$\frac{360 x^{10} - 360 x^{5} + 3 (- (6 x^{5} (6 x^{5}) + 6 x^{5} \cdot - 1 - 1 (6 x^{5} - 1)))}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.7.3

Apply the distributive property.

$\frac{360 x^{10} - 360 x^{5} + 3 (- (6 x^{5} (6 x^{5}) + 6 x^{5} \cdot - 1 - 1 (6 x^{5}) - 1 \cdot - 1))}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.8

Simplify and combine like terms.

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Step 2.31.4.1.8.1

Simplify each term.

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Step 2.31.4.1.8.1.1

Rewrite using the commutative property of multiplication.

$\frac{360 x^{10} - 360 x^{5} + 3 (- (6 \cdot 6 x^{5} x^{5} + 6 x^{5} \cdot - 1 - 1 (6 x^{5}) - 1 \cdot - 1))}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.8.1.2

Multiply $x^{5}$ by $x^{5}$ by adding the exponents.

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Step 2.31.4.1.8.1.2.1

Move $x^{5}$ .

$\frac{360 x^{10} - 360 x^{5} + 3 (- (6 \cdot 6 (x^{5} x^{5}) + 6 x^{5} \cdot - 1 - 1 (6 x^{5}) - 1 \cdot - 1))}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.8.1.2.2

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

$\frac{360 x^{10} - 360 x^{5} + 3 (- (6 \cdot 6 x^{5 + 5} + 6 x^{5} \cdot - 1 - 1 (6 x^{5}) - 1 \cdot - 1))}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.8.1.2.3

Add $5$ and $5$ .

$\frac{360 x^{10} - 360 x^{5} + 3 (- (6 \cdot 6 x^{10} + 6 x^{5} \cdot - 1 - 1 (6 x^{5}) - 1 \cdot - 1))}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.8.1.3

Multiply $6$ by $6$ .

$\frac{360 x^{10} - 360 x^{5} + 3 (- (36 x^{10} + 6 x^{5} \cdot - 1 - 1 (6 x^{5}) - 1 \cdot - 1))}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.8.1.4

Multiply $- 1$ by $6$ .

$\frac{360 x^{10} - 360 x^{5} + 3 (- (36 x^{10} - 6 x^{5} - 1 (6 x^{5}) - 1 \cdot - 1))}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.8.1.5

Multiply $6$ by $- 1$ .

$\frac{360 x^{10} - 360 x^{5} + 3 (- (36 x^{10} - 6 x^{5} - 6 x^{5} - 1 \cdot - 1))}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.8.1.6

Multiply $- 1$ by $- 1$ .

$\frac{360 x^{10} - 360 x^{5} + 3 (- (36 x^{10} - 6 x^{5} - 6 x^{5} + 1))}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.8.2

Subtract $6 x^{5}$ from $- 6 x^{5}$ .

$\frac{360 x^{10} - 360 x^{5} + 3 (- (36 x^{10} - 12 x^{5} + 1))}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.9

Apply the distributive property.

$\frac{360 x^{10} - 360 x^{5} + 3 (- (36 x^{10}) - (- 12 x^{5}) - 1 \cdot 1)}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.10

Simplify.

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Step 2.31.4.1.10.1

Multiply $36$ by $- 1$ .

$\frac{360 x^{10} - 360 x^{5} + 3 (- 36 x^{10} - (- 12 x^{5}) - 1 \cdot 1)}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.10.2

Multiply $- 12$ by $- 1$ .

$\frac{360 x^{10} - 360 x^{5} + 3 (- 36 x^{10} + 12 x^{5} - 1 \cdot 1)}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.10.3

Multiply $- 1$ by $1$ .

$\frac{360 x^{10} - 360 x^{5} + 3 (- 36 x^{10} + 12 x^{5} - 1)}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.11

Apply the distributive property.

$\frac{360 x^{10} - 360 x^{5} + 3 (- 36 x^{10}) + 3 (12 x^{5}) + 3 \cdot - 1}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.12

Simplify.

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Step 2.31.4.1.12.1

Multiply $- 36$ by $3$ .

$\frac{360 x^{10} - 360 x^{5} - 108 x^{10} + 3 (12 x^{5}) + 3 \cdot - 1}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.12.2

Multiply $12$ by $3$ .

$\frac{360 x^{10} - 360 x^{5} - 108 x^{10} + 36 x^{5} + 3 \cdot - 1}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.1.12.3

Multiply $3$ by $- 1$ .

$\frac{360 x^{10} - 360 x^{5} - 108 x^{10} + 36 x^{5} - 3}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.2

Subtract $108 x^{10}$ from $360 x^{10}$ .

$\frac{252 x^{10} - 360 x^{5} + 36 x^{5} - 3}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.4.3

Add $- 360 x^{5}$ and $36 x^{5}$ .

$\frac{252 x^{10} - 324 x^{5} - 3}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.5

Factor $3$ out of $252 x^{10} - 324 x^{5} - 3$ .

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Step 2.31.5.1

Factor $3$ out of $252 x^{10}$ .

$\frac{3 (84 x^{10}) - 324 x^{5} - 3}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.5.2

Factor $3$ out of $- 324 x^{5}$ .

$\frac{3 (84 x^{10}) + 3 (- 108 x^{5}) - 3}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.5.3

Factor $3$ out of $- 3$ .

$\frac{3 (84 x^{10}) + 3 (- 108 x^{5}) + 3 (- 1)}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.5.4

Factor $3$ out of $3 (84 x^{10}) + 3 (- 108 x^{5})$ .

$\frac{3 (84 x^{10} - 108 x^{5}) + 3 (- 1)}{16 {(x^{6} - x)}^{\frac{5}{4}}}$

Step 2.31.5.5

Factor $3$ out of $3 (84 x^{10} - 108 x^{5}) + 3 (- 1)$ .

$f'' (x) = \frac{3 (84 x^{10} - 108 x^{5} - 1)}{16 {(x^{6} - x)}^{\frac{5}{4}}}$