Calculus Examples

$y = {(\frac{x^{2}}{6 x^{3} - 6})}^{5}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ({(\frac{x^{2}}{6 x^{3} - 6})}^{5})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = \frac{x^{2}}{6 x^{3} - 6}$ .

Tap for more steps...

Step 3.1.1

To apply the Chain Rule, set $u$ as $\frac{x^{2}}{6 x^{3} - 6}$ .

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

Step 3.1.2

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

$5 u^{4} \frac{d}{d x} [\frac{x^{2}}{6 x^{3} - 6}]$

Step 3.1.3

Replace all occurrences of $u$ with $\frac{x^{2}}{6 x^{3} - 6}$ .

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

Step 3.2

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 $f (x) = x^{2}$ and $g (x) = 6 x^{3} - 6$ .

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

Step 3.3

Differentiate.

Tap for more steps...

Step 3.3.1

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

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

Step 3.3.2

Move $2$ to the left of $6 x^{3} - 6$ .

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

Step 3.3.3

By the Sum Rule, the derivative of $6 x^{3} - 6$ with respect to $x$ is $\frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [- 6]$ .

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

Step 3.3.4

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{3}$ with respect to $x$ is $6 \frac{d}{d x} [x^{3}]$ .

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

Step 3.3.5

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

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

Step 3.3.6

Multiply $3$ by $6$ .

$5 {(\frac{x^{2}}{6 x^{3} - 6})}^{4} \frac{2 (6 x^{3} - 6) x - x^{2} (18 x^{2} + \frac{d}{d x} [- 6])}{{(6 x^{3} - 6)}^{2}}$

Step 3.3.7

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$5 {(\frac{x^{2}}{6 x^{3} - 6})}^{4} \frac{2 (6 x^{3} - 6) x - x^{2} (18 x^{2} + 0)}{{(6 x^{3} - 6)}^{2}}$

Step 3.3.8

Simplify the expression.

Tap for more steps...

Step 3.3.8.1

Add $18 x^{2}$ and $0$ .

$5 {(\frac{x^{2}}{6 x^{3} - 6})}^{4} \frac{2 (6 x^{3} - 6) x - x^{2} (18 x^{2})}{{(6 x^{3} - 6)}^{2}}$

Step 3.3.8.2

Multiply $18$ by $- 1$ .

$5 {(\frac{x^{2}}{6 x^{3} - 6})}^{4} \frac{2 (6 x^{3} - 6) x - 18 x^{2} x^{2}}{{(6 x^{3} - 6)}^{2}}$

Step 3.4

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 3.4.1

Move $x^{2}$ .

$5 {(\frac{x^{2}}{6 x^{3} - 6})}^{4} \frac{2 (6 x^{3} - 6) x - 18 (x^{2} x^{2})}{{(6 x^{3} - 6)}^{2}}$

Step 3.4.2

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

$5 {(\frac{x^{2}}{6 x^{3} - 6})}^{4} \frac{2 (6 x^{3} - 6) x - 18 x^{2 + 2}}{{(6 x^{3} - 6)}^{2}}$

Step 3.4.3

Add $2$ and $2$ .

$5 {(\frac{x^{2}}{6 x^{3} - 6})}^{4} \frac{2 (6 x^{3} - 6) x - 18 x^{4}}{{(6 x^{3} - 6)}^{2}}$

Step 3.5

Combine $\frac{2 (6 x^{3} - 6) x - 18 x^{4}}{{(6 x^{3} - 6)}^{2}}$ and $5$ .

$\frac{(2 (6 x^{3} - 6) x - 18 x^{4}) \cdot 5}{{(6 x^{3} - 6)}^{2}} {(\frac{x^{2}}{6 x^{3} - 6})}^{4}$

Step 3.6

Move $5$ to the left of $2 (6 x^{3} - 6) x - 18 x^{4}$ .

$\frac{5 \cdot (2 (6 x^{3} - 6) x - 18 x^{4})}{{(6 x^{3} - 6)}^{2}} {(\frac{x^{2}}{6 x^{3} - 6})}^{4}$

Step 3.7

Simplify.

Tap for more steps...

Step 3.7.1

Apply the product rule to $\frac{x^{2}}{6 x^{3} - 6}$ .

$\frac{5 (2 (6 x^{3} - 6) x - 18 x^{4})}{{(6 x^{3} - 6)}^{2}} \cdot \frac{{(x^{2})}^{4}}{{(6 x^{3} - 6)}^{4}}$

Step 3.7.2

Apply the distributive property.

$\frac{5 ((2 (6 x^{3}) + 2 \cdot - 6) x - 18 x^{4})}{{(6 x^{3} - 6)}^{2}} \cdot \frac{{(x^{2})}^{4}}{{(6 x^{3} - 6)}^{4}}$

Step 3.7.3

Apply the distributive property.

$\frac{5 (2 (6 x^{3}) x + 2 \cdot - 6 x - 18 x^{4})}{{(6 x^{3} - 6)}^{2}} \cdot \frac{{(x^{2})}^{4}}{{(6 x^{3} - 6)}^{4}}$

Step 3.7.4

Apply the distributive property.

$\frac{5 (2 (6 x^{3}) x) + 5 (2 \cdot - 6 x) + 5 (- 18 x^{4})}{{(6 x^{3} - 6)}^{2}} \cdot \frac{{(x^{2})}^{4}}{{(6 x^{3} - 6)}^{4}}$

Step 3.7.5

Combine terms.

Tap for more steps...

Step 3.7.5.1

Multiply $6$ by $2$ .

$\frac{5 (12 x^{3} x) + 5 (2 \cdot - 6 x) + 5 (- 18 x^{4})}{{(6 x^{3} - 6)}^{2}} \cdot \frac{{(x^{2})}^{4}}{{(6 x^{3} - 6)}^{4}}$

Step 3.7.5.2

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

$\frac{5 (12 (x^{1} x^{3})) + 5 (2 \cdot - 6 x) + 5 (- 18 x^{4})}{{(6 x^{3} - 6)}^{2}} \cdot \frac{{(x^{2})}^{4}}{{(6 x^{3} - 6)}^{4}}$

Step 3.7.5.3

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

$\frac{5 (12 x^{1 + 3}) + 5 (2 \cdot - 6 x) + 5 (- 18 x^{4})}{{(6 x^{3} - 6)}^{2}} \cdot \frac{{(x^{2})}^{4}}{{(6 x^{3} - 6)}^{4}}$

Step 3.7.5.4

Add $1$ and $3$ .

$\frac{5 (12 x^{4}) + 5 (2 \cdot - 6 x) + 5 (- 18 x^{4})}{{(6 x^{3} - 6)}^{2}} \cdot \frac{{(x^{2})}^{4}}{{(6 x^{3} - 6)}^{4}}$

Step 3.7.5.5

Multiply $12$ by $5$ .

$\frac{60 x^{4} + 5 (2 \cdot - 6 x) + 5 (- 18 x^{4})}{{(6 x^{3} - 6)}^{2}} \cdot \frac{{(x^{2})}^{4}}{{(6 x^{3} - 6)}^{4}}$

Step 3.7.5.6

Multiply $2$ by $- 6$ .

$\frac{60 x^{4} + 5 (- 12 x) + 5 (- 18 x^{4})}{{(6 x^{3} - 6)}^{2}} \cdot \frac{{(x^{2})}^{4}}{{(6 x^{3} - 6)}^{4}}$

Step 3.7.5.7

Multiply $- 12$ by $5$ .

$\frac{60 x^{4} - 60 x + 5 (- 18 x^{4})}{{(6 x^{3} - 6)}^{2}} \cdot \frac{{(x^{2})}^{4}}{{(6 x^{3} - 6)}^{4}}$

Step 3.7.5.8

Multiply $- 18$ by $5$ .

$\frac{60 x^{4} - 60 x - 90 x^{4}}{{(6 x^{3} - 6)}^{2}} \cdot \frac{{(x^{2})}^{4}}{{(6 x^{3} - 6)}^{4}}$

Step 3.7.5.9

Subtract $90 x^{4}$ from $60 x^{4}$ .

$\frac{- 30 x^{4} - 60 x}{{(6 x^{3} - 6)}^{2}} \cdot \frac{{(x^{2})}^{4}}{{(6 x^{3} - 6)}^{4}}$

Step 3.7.5.10

Multiply the exponents in ${(x^{2})}^{4}$ .

Tap for more steps...

Step 3.7.5.10.1

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

$\frac{- 30 x^{4} - 60 x}{{(6 x^{3} - 6)}^{2}} \cdot \frac{x^{2 \cdot 4}}{{(6 x^{3} - 6)}^{4}}$

Step 3.7.5.10.2

Multiply $2$ by $4$ .

$\frac{- 30 x^{4} - 60 x}{{(6 x^{3} - 6)}^{2}} \cdot \frac{x^{8}}{{(6 x^{3} - 6)}^{4}}$

Step 3.7.5.11

Multiply $\frac{- 30 x^{4} - 60 x}{{(6 x^{3} - 6)}^{2}}$ by $\frac{x^{8}}{{(6 x^{3} - 6)}^{4}}$ .

$\frac{(- 30 x^{4} - 60 x) x^{8}}{{(6 x^{3} - 6)}^{2} {(6 x^{3} - 6)}^{4}}$

Step 3.7.5.12

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

Tap for more steps...

Step 3.7.5.12.1

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

$\frac{(- 30 x^{4} - 60 x) x^{8}}{{(6 x^{3} - 6)}^{2 + 4}}$

Step 3.7.5.12.2

Add $2$ and $4$ .

$\frac{(- 30 x^{4} - 60 x) x^{8}}{{(6 x^{3} - 6)}^{6}}$

Step 3.7.6

Reorder terms.

$\frac{x^{8} (- 30 x^{4} - 60 x)}{{(6 x^{3} - 6)}^{6}}$

Step 3.7.7

Simplify the numerator.

Tap for more steps...

Step 3.7.7.1

Factor $30 x$ out of $- 30 x^{4} - 60 x$ .

Tap for more steps...

Step 3.7.7.1.1

Factor $30 x$ out of $- 30 x^{4}$ .

$\frac{x^{8} (30 x (- x^{3}) - 60 x)}{{(6 x^{3} - 6)}^{6}}$

Step 3.7.7.1.2

Factor $30 x$ out of $- 60 x$ .

$\frac{x^{8} (30 x (- x^{3}) + 30 x (- 2))}{{(6 x^{3} - 6)}^{6}}$

Step 3.7.7.1.3

Factor $30 x$ out of $30 x (- x^{3}) + 30 x (- 2)$ .

$\frac{x^{8} (30 x (- x^{3} - 2))}{{(6 x^{3} - 6)}^{6}}$

$\frac{x^{8} \cdot 30 x (- x^{3} - 2)}{{(6 x^{3} - 6)}^{6}}$

Step 3.7.7.2

Combine exponents.

Tap for more steps...

Step 3.7.7.2.1

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

$\frac{30 (x^{8} x^{1}) (- x^{3} - 2)}{{(6 x^{3} - 6)}^{6}}$

Step 3.7.7.2.2

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

$\frac{30 x^{8 + 1} (- x^{3} - 2)}{{(6 x^{3} - 6)}^{6}}$

Step 3.7.7.2.3

Add $8$ and $1$ .

$\frac{30 x^{9} (- x^{3} - 2)}{{(6 x^{3} - 6)}^{6}}$

Step 3.7.8

Simplify the denominator.

Tap for more steps...

Step 3.7.8.1

Factor $6$ out of $6 x^{3} - 6$ .

Tap for more steps...

Step 3.7.8.1.1

Factor $6$ out of $6 x^{3}$ .

$\frac{30 x^{9} (- x^{3} - 2)}{{(6 (x^{3}) - 6)}^{6}}$

Step 3.7.8.1.2

Factor $6$ out of $- 6$ .

$\frac{30 x^{9} (- x^{3} - 2)}{{(6 (x^{3}) + 6 (- 1))}^{6}}$

Step 3.7.8.1.3

Factor $6$ out of $6 (x^{3}) + 6 (- 1)$ .

$\frac{30 x^{9} (- x^{3} - 2)}{{(6 (x^{3} - 1))}^{6}}$

Step 3.7.8.2

Rewrite $1$ as $1^{3}$ .

$\frac{30 x^{9} (- x^{3} - 2)}{{(6 (x^{3} - 1^{3}))}^{6}}$

Step 3.7.8.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 1$ .

$\frac{30 x^{9} (- x^{3} - 2)}{{(6 ((x - 1) (x^{2} + x \cdot 1 + 1^{2})))}^{6}}$

Step 3.7.8.4

Simplify.

Tap for more steps...

Step 3.7.8.4.1

Multiply $x$ by $1$ .

$\frac{30 x^{9} (- x^{3} - 2)}{{(6 ((x - 1) (x^{2} + x + 1^{2})))}^{6}}$

Step 3.7.8.4.2

One to any power is one.

$\frac{30 x^{9} (- x^{3} - 2)}{{(6 ((x - 1) (x^{2} + x + 1)))}^{6}}$

$\frac{30 x^{9} (- x^{3} - 2)}{{(6 (x - 1) (x^{2} + x + 1))}^{6}}$

Step 3.7.8.5

Apply the product rule to $6 (x - 1) (x^{2} + x + 1)$ .

$\frac{30 x^{9} (- x^{3} - 2)}{{(6 (x - 1))}^{6} {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.6

Apply the distributive property.

$\frac{30 x^{9} (- x^{3} - 2)}{{(6 x + 6 \cdot - 1)}^{6} {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.7

Multiply $6$ by $- 1$ .

$\frac{30 x^{9} (- x^{3} - 2)}{{(6 x - 6)}^{6} {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.8

Use the Binomial Theorem.

$\frac{30 x^{9} (- x^{3} - 2)}{({(6 x)}^{6} + 6 {(6 x)}^{5} \cdot - 6 + 15 {(6 x)}^{4} {(- 6)}^{2} + 20 {(6 x)}^{3} {(- 6)}^{3} + 15 {(6 x)}^{2} {(- 6)}^{4} + 6 (6 x) {(- 6)}^{5} + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9

Simplify each term.

Tap for more steps...

Step 3.7.8.9.1

Apply the product rule to $6 x$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(6^{6} x^{6} + 6 {(6 x)}^{5} \cdot - 6 + 15 {(6 x)}^{4} {(- 6)}^{2} + 20 {(6 x)}^{3} {(- 6)}^{3} + 15 {(6 x)}^{2} {(- 6)}^{4} + 6 (6 x) {(- 6)}^{5} + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.2

Raise $6$ to the power of $6$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} + 6 {(6 x)}^{5} \cdot - 6 + 15 {(6 x)}^{4} {(- 6)}^{2} + 20 {(6 x)}^{3} {(- 6)}^{3} + 15 {(6 x)}^{2} {(- 6)}^{4} + 6 (6 x) {(- 6)}^{5} + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.3

Apply the product rule to $6 x$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} + 6 (6^{5} x^{5}) \cdot - 6 + 15 {(6 x)}^{4} {(- 6)}^{2} + 20 {(6 x)}^{3} {(- 6)}^{3} + 15 {(6 x)}^{2} {(- 6)}^{4} + 6 (6 x) {(- 6)}^{5} + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.4

Multiply $6$ by $6^{5}$ by adding the exponents.

Tap for more steps...

Step 3.7.8.9.4.1

Move $6^{5}$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} + 6^{5} \cdot 6 x^{5} \cdot - 6 + 15 {(6 x)}^{4} {(- 6)}^{2} + 20 {(6 x)}^{3} {(- 6)}^{3} + 15 {(6 x)}^{2} {(- 6)}^{4} + 6 (6 x) {(- 6)}^{5} + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.4.2

Multiply $6^{5}$ by $6$ .

Tap for more steps...

Step 3.7.8.9.4.2.1

Raise $6$ to the power of $1$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} + 6^{5} \cdot 6^{1} x^{5} \cdot - 6 + 15 {(6 x)}^{4} {(- 6)}^{2} + 20 {(6 x)}^{3} {(- 6)}^{3} + 15 {(6 x)}^{2} {(- 6)}^{4} + 6 (6 x) {(- 6)}^{5} + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.4.2.2

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

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} + 6^{5 + 1} x^{5} \cdot - 6 + 15 {(6 x)}^{4} {(- 6)}^{2} + 20 {(6 x)}^{3} {(- 6)}^{3} + 15 {(6 x)}^{2} {(- 6)}^{4} + 6 (6 x) {(- 6)}^{5} + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.4.3

Add $5$ and $1$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} + 6^{6} x^{5} \cdot - 6 + 15 {(6 x)}^{4} {(- 6)}^{2} + 20 {(6 x)}^{3} {(- 6)}^{3} + 15 {(6 x)}^{2} {(- 6)}^{4} + 6 (6 x) {(- 6)}^{5} + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.5

Raise $6$ to the power of $6$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} + 46656 x^{5} \cdot - 6 + 15 {(6 x)}^{4} {(- 6)}^{2} + 20 {(6 x)}^{3} {(- 6)}^{3} + 15 {(6 x)}^{2} {(- 6)}^{4} + 6 (6 x) {(- 6)}^{5} + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.6

Multiply $- 6$ by $46656$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} - 279936 x^{5} + 15 {(6 x)}^{4} {(- 6)}^{2} + 20 {(6 x)}^{3} {(- 6)}^{3} + 15 {(6 x)}^{2} {(- 6)}^{4} + 6 (6 x) {(- 6)}^{5} + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.7

Apply the product rule to $6 x$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} - 279936 x^{5} + 15 (6^{4} x^{4}) {(- 6)}^{2} + 20 {(6 x)}^{3} {(- 6)}^{3} + 15 {(6 x)}^{2} {(- 6)}^{4} + 6 (6 x) {(- 6)}^{5} + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.8

Raise $6$ to the power of $4$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} - 279936 x^{5} + 15 (1296 x^{4}) {(- 6)}^{2} + 20 {(6 x)}^{3} {(- 6)}^{3} + 15 {(6 x)}^{2} {(- 6)}^{4} + 6 (6 x) {(- 6)}^{5} + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.9

Multiply $1296$ by $15$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} - 279936 x^{5} + 19440 x^{4} {(- 6)}^{2} + 20 {(6 x)}^{3} {(- 6)}^{3} + 15 {(6 x)}^{2} {(- 6)}^{4} + 6 (6 x) {(- 6)}^{5} + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.10

Raise $- 6$ to the power of $2$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} - 279936 x^{5} + 19440 x^{4} \cdot 36 + 20 {(6 x)}^{3} {(- 6)}^{3} + 15 {(6 x)}^{2} {(- 6)}^{4} + 6 (6 x) {(- 6)}^{5} + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.11

Multiply $36$ by $19440$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} - 279936 x^{5} + 699840 x^{4} + 20 {(6 x)}^{3} {(- 6)}^{3} + 15 {(6 x)}^{2} {(- 6)}^{4} + 6 (6 x) {(- 6)}^{5} + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.12

Apply the product rule to $6 x$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} - 279936 x^{5} + 699840 x^{4} + 20 (6^{3} x^{3}) {(- 6)}^{3} + 15 {(6 x)}^{2} {(- 6)}^{4} + 6 (6 x) {(- 6)}^{5} + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.13

Raise $6$ to the power of $3$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} - 279936 x^{5} + 699840 x^{4} + 20 (216 x^{3}) {(- 6)}^{3} + 15 {(6 x)}^{2} {(- 6)}^{4} + 6 (6 x) {(- 6)}^{5} + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.14

Multiply $216$ by $20$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} - 279936 x^{5} + 699840 x^{4} + 4320 x^{3} {(- 6)}^{3} + 15 {(6 x)}^{2} {(- 6)}^{4} + 6 (6 x) {(- 6)}^{5} + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.15

Raise $- 6$ to the power of $3$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} - 279936 x^{5} + 699840 x^{4} + 4320 x^{3} \cdot - 216 + 15 {(6 x)}^{2} {(- 6)}^{4} + 6 (6 x) {(- 6)}^{5} + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.16

Multiply $- 216$ by $4320$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} - 279936 x^{5} + 699840 x^{4} - 933120 x^{3} + 15 {(6 x)}^{2} {(- 6)}^{4} + 6 (6 x) {(- 6)}^{5} + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.17

Apply the product rule to $6 x$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} - 279936 x^{5} + 699840 x^{4} - 933120 x^{3} + 15 (6^{2} x^{2}) {(- 6)}^{4} + 6 (6 x) {(- 6)}^{5} + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.18

Raise $6$ to the power of $2$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} - 279936 x^{5} + 699840 x^{4} - 933120 x^{3} + 15 (36 x^{2}) {(- 6)}^{4} + 6 (6 x) {(- 6)}^{5} + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.19

Multiply $36$ by $15$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} - 279936 x^{5} + 699840 x^{4} - 933120 x^{3} + 540 x^{2} {(- 6)}^{4} + 6 (6 x) {(- 6)}^{5} + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.20

Raise $- 6$ to the power of $4$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} - 279936 x^{5} + 699840 x^{4} - 933120 x^{3} + 540 x^{2} \cdot 1296 + 6 (6 x) {(- 6)}^{5} + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.21

Multiply $1296$ by $540$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} - 279936 x^{5} + 699840 x^{4} - 933120 x^{3} + 699840 x^{2} + 6 (6 x) {(- 6)}^{5} + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.22

Multiply $6$ by $6$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} - 279936 x^{5} + 699840 x^{4} - 933120 x^{3} + 699840 x^{2} + 36 x {(- 6)}^{5} + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.23

Raise $- 6$ to the power of $5$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} - 279936 x^{5} + 699840 x^{4} - 933120 x^{3} + 699840 x^{2} + 36 x \cdot - 7776 + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.24

Multiply $- 7776$ by $36$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} - 279936 x^{5} + 699840 x^{4} - 933120 x^{3} + 699840 x^{2} - 279936 x + {(- 6)}^{6}) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.9.25

Raise $- 6$ to the power of $6$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 x^{6} - 279936 x^{5} + 699840 x^{4} - 933120 x^{3} + 699840 x^{2} - 279936 x + 46656) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.10

Factor $46656$ out of $46656 x^{6} - 279936 x^{5} + 699840 x^{4} - 933120 x^{3} + 699840 x^{2} - 279936 x + 46656$ .

Tap for more steps...

Step 3.7.8.10.1

Factor $46656$ out of $46656 x^{6}$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 (x^{6}) - 279936 x^{5} + 699840 x^{4} - 933120 x^{3} + 699840 x^{2} - 279936 x + 46656) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.10.2

Factor $46656$ out of $- 279936 x^{5}$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 (x^{6}) + 46656 (- 6 x^{5}) + 699840 x^{4} - 933120 x^{3} + 699840 x^{2} - 279936 x + 46656) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.10.3

Factor $46656$ out of $699840 x^{4}$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 (x^{6}) + 46656 (- 6 x^{5}) + 46656 (15 x^{4}) - 933120 x^{3} + 699840 x^{2} - 279936 x + 46656) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.10.4

Factor $46656$ out of $- 933120 x^{3}$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 (x^{6}) + 46656 (- 6 x^{5}) + 46656 (15 x^{4}) + 46656 (- 20 x^{3}) + 699840 x^{2} - 279936 x + 46656) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.10.5

Factor $46656$ out of $699840 x^{2}$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 (x^{6}) + 46656 (- 6 x^{5}) + 46656 (15 x^{4}) + 46656 (- 20 x^{3}) + 46656 (15 x^{2}) - 279936 x + 46656) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.10.6

Factor $46656$ out of $- 279936 x$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 (x^{6}) + 46656 (- 6 x^{5}) + 46656 (15 x^{4}) + 46656 (- 20 x^{3}) + 46656 (15 x^{2}) + 46656 (- 6 x) + 46656) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.10.7

Factor $46656$ out of $46656$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 (x^{6}) + 46656 (- 6 x^{5}) + 46656 (15 x^{4}) + 46656 (- 20 x^{3}) + 46656 (15 x^{2}) + 46656 (- 6 x) + 46656 (1)) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.10.8

Factor $46656$ out of $46656 (x^{6}) + 46656 (- 6 x^{5})$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 (x^{6} - 6 x^{5}) + 46656 (15 x^{4}) + 46656 (- 20 x^{3}) + 46656 (15 x^{2}) + 46656 (- 6 x) + 46656 (1)) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.10.9

Factor $46656$ out of $46656 (x^{6} - 6 x^{5}) + 46656 (15 x^{4})$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 (x^{6} - 6 x^{5} + 15 x^{4}) + 46656 (- 20 x^{3}) + 46656 (15 x^{2}) + 46656 (- 6 x) + 46656 (1)) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.10.10

Factor $46656$ out of $46656 (x^{6} - 6 x^{5} + 15 x^{4}) + 46656 (- 20 x^{3})$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 (x^{6} - 6 x^{5} + 15 x^{4} - 20 x^{3}) + 46656 (15 x^{2}) + 46656 (- 6 x) + 46656 (1)) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.10.11

Factor $46656$ out of $46656 (x^{6} - 6 x^{5} + 15 x^{4} - 20 x^{3}) + 46656 (15 x^{2})$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 (x^{6} - 6 x^{5} + 15 x^{4} - 20 x^{3} + 15 x^{2}) + 46656 (- 6 x) + 46656 (1)) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.10.12

Factor $46656$ out of $46656 (x^{6} - 6 x^{5} + 15 x^{4} - 20 x^{3} + 15 x^{2}) + 46656 (- 6 x)$ .

$\frac{30 x^{9} (- x^{3} - 2)}{(46656 (x^{6} - 6 x^{5} + 15 x^{4} - 20 x^{3} + 15 x^{2} - 6 x) + 46656 (1)) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.10.13

Factor $46656$ out of $46656 (x^{6} - 6 x^{5} + 15 x^{4} - 20 x^{3} + 15 x^{2} - 6 x) + 46656 (1)$ .

$\frac{30 x^{9} (- x^{3} - 2)}{46656 (x^{6} - 6 x^{5} + 15 x^{4} - 20 x^{3} + 15 x^{2} - 6 x + 1) {(x^{2} + x + 1)}^{6}}$

Step 3.7.8.11

Factor $x^{6} - 6 x^{5} + 15 x^{4} - 20 x^{3} + 15 x^{2} - 6 x + 1$ using the binomial theorem.

$\frac{30 x^{9} (- x^{3} - 2)}{46656 {(1 - x)}^{6} {(x^{2} + x + 1)}^{6}}$

Step 3.7.9

Cancel the common factor of $30$ and $46656$ .

Tap for more steps...

Step 3.7.9.1

Factor $6$ out of $30 x^{9} (- x^{3} - 2)$ .

$\frac{6 (5 x^{9} (- x^{3} - 2))}{46656 {(1 - x)}^{6} {(x^{2} + x + 1)}^{6}}$

Step 3.7.9.2

Cancel the common factors.

Tap for more steps...

Step 3.7.9.2.1

Factor $6$ out of $46656 {(1 - x)}^{6} {(x^{2} + x + 1)}^{6}$ .

$\frac{6 (5 x^{9} (- x^{3} - 2))}{6 (7776 {(1 - x)}^{6} {(x^{2} + x + 1)}^{6})}$

Step 3.7.9.2.2

Cancel the common factor.

$\frac{6 (5 x^{9} (- x^{3} - 2))}{6 (7776 {(1 - x)}^{6} {(x^{2} + x + 1)}^{6})}$

Step 3.7.9.2.3

Rewrite the expression.

$\frac{5 x^{9} (- x^{3} - 2)}{7776 {(1 - x)}^{6} {(x^{2} + x + 1)}^{6}}$

Step 3.7.10

Factor $- 1$ out of $- x^{3}$ .

$\frac{5 x^{9} (- (x^{3}) - 2)}{7776 {(1 - x)}^{6} {(x^{2} + x + 1)}^{6}}$

Step 3.7.11

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{5 x^{9} (- (x^{3}) - 1 (2))}{7776 {(1 - x)}^{6} {(x^{2} + x + 1)}^{6}}$

Step 3.7.12

Factor $- 1$ out of $- (x^{3}) - 1 (2)$ .

$\frac{5 x^{9} (- (x^{3} + 2))}{7776 {(1 - x)}^{6} {(x^{2} + x + 1)}^{6}}$

Step 3.7.13

Rewrite $- (x^{3} + 2)$ as $- 1 (x^{3} + 2)$ .

$\frac{5 x^{9} (- 1 (x^{3} + 2))}{7776 {(1 - x)}^{6} {(x^{2} + x + 1)}^{6}}$

Step 3.7.14

Move the negative in front of the fraction.

$- \frac{(5 x^{9}) (x^{3} + 2)}{7776 {(1 - x)}^{6} {(x^{2} + x + 1)}^{6}}$

Step 3.7.15

Reorder factors in $- \frac{(5 x^{9}) (x^{3} + 2)}{7776 {(1 - x)}^{6} {(x^{2} + x + 1)}^{6}}$ .

$- \frac{5 x^{9} (x^{3} + 2)}{7776 {(1 - x)}^{6} {(x^{2} + x + 1)}^{6}}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = - \frac{5 x^{9} (x^{3} + 2)}{7776 {(1 - x)}^{6} {(x^{2} + x + 1)}^{6}}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{5 x^{9} (x^{3} + 2)}{7776 {(1 - x)}^{6} {(x^{2} + x + 1)}^{6}}$