Find dx/dy y=arccos(8x^5)

$y = \arccos (8 x^{5})$

Differentiate both sides of the equation.

$\frac{d}{d y} (y) = \frac{d}{d y} (\arccos (8 x^{5}))$

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

$1$

Differentiate the right side of the equation.

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = \arccos (y)$ and $g (y) = 8 x^{5}$ .

To apply the Chain Rule, set $u_{1}$ as $8 x^{5}$ .

$\frac{d}{d u_{1}} [\arccos (u_{1})] \frac{d}{d y} [8 x^{5}]$

The derivative of $\arccos (u_{1})$ with respect to $u_{1}$ is $- \frac{1}{\sqrt{1 - {u_{1}}^{2}}}$ .

$- \frac{1}{\sqrt{1 - {u_{1}}^{2}}} \frac{d}{d y} [8 x^{5}]$

Replace all occurrences of $u_{1}$ with $8 x^{5}$ .

$- \frac{1}{\sqrt{1 - {(8 x^{5})}^{2}}} \frac{d}{d y} [8 x^{5}]$

Differentiate using the Constant Multiple Rule.

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

$- \frac{1}{\sqrt{1 - {(8 (x^{5}))}^{2}}} \frac{d}{d y} [8 x^{5}]$

Simplify the expression.

Apply the product rule to $8 (x^{5})$ .

$- \frac{1}{\sqrt{1 - (8^{2} {(x^{5})}^{2})}} \frac{d}{d y} [8 x^{5}]$

Raise $8$ to the power of $2$ .

$- \frac{1}{\sqrt{1 - (64 {(x^{5})}^{2})}} \frac{d}{d y} [8 x^{5}]$

Multiply the exponents in ${(x^{5})}^{2}$ .

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

$- \frac{1}{\sqrt{1 - (64 x^{5 \cdot 2})}} \frac{d}{d y} [8 x^{5}]$

Multiply $5$ by $2$ .

$- \frac{1}{\sqrt{1 - (64 x^{10})}} \frac{d}{d y} [8 x^{5}]$

Multiply $64$ by $- 1$ .

$- \frac{1}{\sqrt{1 - 64 x^{10}}} \frac{d}{d y} [8 x^{5}]$

Since $8$ is constant with respect to $y$ , the derivative of $8 x^{5}$ with respect to $y$ is $8 \frac{d}{d y} [x^{5}]$ .

$- \frac{1}{\sqrt{1 - 64 x^{10}}} (8 \frac{d}{d y} [x^{5}])$

Combine fractions.

Multiply $8$ by $- 1$ .

$- 8 \frac{1}{\sqrt{1 - 64 x^{10}}} \frac{d}{d y} [x^{5}]$

Combine $- 8$ and $\frac{1}{\sqrt{1 - 64 x^{10}}}$ .

$\frac{- 8}{\sqrt{1 - 64 x^{10}}} \frac{d}{d y} [x^{5}]$

Move the negative in front of the fraction.

$- \frac{8}{\sqrt{1 - 64 x^{10}}} \frac{d}{d y} [x^{5}]$

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

To apply the Chain Rule, set $u_{2}$ as $x$ .

$- \frac{8}{\sqrt{1 - 64 x^{10}}} (\frac{d}{d u_{2}} [{u_{2}}^{5}] \frac{d}{d y} [x])$

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 5$ .

$- \frac{8}{\sqrt{1 - 64 x^{10}}} (5 {u_{2}}^{4} \frac{d}{d y} [x])$

Replace all occurrences of $u_{2}$ with $x$ .

$- \frac{8}{\sqrt{1 - 64 x^{10}}} (5 x^{4} \frac{d}{d y} [x])$

Combine fractions.

Multiply $5$ by $- 1$ .

$- 5 \frac{8}{\sqrt{1 - 64 x^{10}}} (x^{4} \frac{d}{d y} [x])$

Combine $- 5$ and $\frac{8}{\sqrt{1 - 64 x^{10}}}$ .

$\frac{- 5 \cdot 8}{\sqrt{1 - 64 x^{10}}} (x^{4} \frac{d}{d y} [x])$

Multiply $- 5$ by $8$ .

$\frac{- 40}{\sqrt{1 - 64 x^{10}}} (x^{4} \frac{d}{d y} [x])$

Combine $x^{4}$ and $\frac{- 40}{\sqrt{1 - 64 x^{10}}}$ .

$\frac{x^{4} \cdot - 40}{\sqrt{1 - 64 x^{10}}} \frac{d}{d y} [x]$

Simplify the expression.

Move $- 40$ to the left of $x^{4}$ .

$\frac{- 40 \cdot x^{4}}{\sqrt{1 - 64 x^{10}}} \frac{d}{d y} [x]$

Move the negative in front of the fraction.

$- \frac{(40) x^{4}}{\sqrt{1 - 64 x^{10}}} \frac{d}{d y} [x]$

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

$- \frac{40 x^{4}}{\sqrt{1 - 64 x^{10}}} x'$

Combine $x'$ and $\frac{40 x^{4}}{\sqrt{1 - 64 x^{10}}}$ .

$- \frac{x' (40 x^{4})}{\sqrt{1 - 64 x^{10}}}$

Simplify the numerator.

Rewrite using the commutative property of multiplication.

$- \frac{40 x' x^{4}}{\sqrt{1 - 64 x^{10}}}$

Reorder factors in $40 x' x^{4}$ .

$- \frac{40 x^{4} x'}{\sqrt{1 - 64 x^{10}}}$

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

$1 = - \frac{40 x^{4} x'}{\sqrt{1 - 64 x^{10}}}$

Solve for $x'$ .

Rewrite the equation as $- \frac{40 x^{4} x'}{\sqrt{1 - 64 x^{10}}} = 1$ .

$- \frac{40 x^{4} x'}{\sqrt{1 - 64 x^{10}}} = 1$

Divide each term in $- \frac{40 x^{4} x'}{\sqrt{1 - 64 x^{10}}} = 1$ by $- 1$ and simplify.

Divide each term in $- \frac{40 x^{4} x'}{\sqrt{1 - 64 x^{10}}} = 1$ by $- 1$ .

$\frac{- \frac{40 x^{4} x'}{\sqrt{1 - 64 x^{10}}}}{- 1} = \frac{1}{- 1}$

Simplify the left side.

Reduce the expression by cancelling the common factors.

Dividing two negative values results in a positive value.

$\frac{\frac{40 x^{4} x'}{\sqrt{1 - 64 x^{10}}}}{1} = \frac{1}{- 1}$

Divide $\frac{40 x^{4} x'}{\sqrt{1 - 64 x^{10}}}$ by $1$ .

$\frac{40 x^{4} x'}{\sqrt{1 - 64 x^{10}}} = \frac{1}{- 1}$

Simplify the denominator.

Rewrite $1$ as $1^{2}$ .

$\frac{40 x^{4} x'}{\sqrt{1^{2} - 64 x^{10}}} = \frac{1}{- 1}$

Rewrite $64 x^{10}$ as ${(8 x^{5})}^{2}$ .

$\frac{40 x^{4} x'}{\sqrt{1^{2} - {(8 x^{5})}^{2}}} = \frac{1}{- 1}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = 8 x^{5}$ .

$\frac{40 x^{4} x'}{\sqrt{(1 + 8 x^{5}) (1 - (8 x^{5}))}} = \frac{1}{- 1}$

Multiply $8$ by $- 1$ .

$\frac{40 x^{4} x'}{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}} = \frac{1}{- 1}$

Multiply $\frac{40 x^{4} x'}{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$ by $\frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$ .

$\frac{40 x^{4} x'}{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}} \cdot \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}} = \frac{1}{- 1}$

Combine and simplify the denominator.

Multiply $\frac{40 x^{4} x'}{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$ by $\frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$ .

$\frac{40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}} = \frac{1}{- 1}$

Raise $\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}$ to the power of $1$ .

$\frac{40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}^{1} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}} = \frac{1}{- 1}$

Raise $\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}$ to the power of $1$ .

$\frac{40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}^{1} {\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}^{1}} = \frac{1}{- 1}$

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

$\frac{40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}^{1 + 1}} = \frac{1}{- 1}$

Add $1$ and $1$ .

$\frac{40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}^{2}} = \frac{1}{- 1}$

Rewrite ${\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}^{2}$ as $(1 + 8 x^{5}) (1 - 8 x^{5})$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}$ as ${((1 + 8 x^{5}) (1 - 8 x^{5}))}^{\frac{1}{2}}$ .

$\frac{40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{{({((1 + 8 x^{5}) (1 - 8 x^{5}))}^{\frac{1}{2}})}^{2}} = \frac{1}{- 1}$

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

$\frac{40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{{((1 + 8 x^{5}) (1 - 8 x^{5}))}^{\frac{1}{2} \cdot 2}} = \frac{1}{- 1}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{{((1 + 8 x^{5}) (1 - 8 x^{5}))}^{\frac{2}{2}}} = \frac{1}{- 1}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{{((1 + 8 x^{5}) (1 - 8 x^{5}))}^{\frac{2}{2}}} = \frac{1}{- 1}$

Rewrite the expression.

$\frac{40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{{((1 + 8 x^{5}) (1 - 8 x^{5}))}^{1}} = \frac{1}{- 1}$

Simplify.

$\frac{40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{(1 + 8 x^{5}) (1 - 8 x^{5})} = \frac{1}{- 1}$

Simplify the right side.

Divide $1$ by $- 1$ .

$\frac{40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{(1 + 8 x^{5}) (1 - 8 x^{5})} = - 1$

Multiply both sides by $(1 + 8 x^{5}) (1 - 8 x^{5})$ .

$\frac{40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{(1 + 8 x^{5}) (1 - 8 x^{5})} ((1 + 8 x^{5}) (1 - 8 x^{5})) = - ((1 + 8 x^{5}) (1 - 8 x^{5}))$

Simplify.

Simplify the left side.

Cancel the common factor of $(1 + 8 x^{5}) (1 - 8 x^{5})$ .

Cancel the common factor.

$\frac{40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{(1 + 8 x^{5}) (1 - 8 x^{5})} ((1 + 8 x^{5}) (1 - 8 x^{5})) = - ((1 + 8 x^{5}) (1 - 8 x^{5}))$

Rewrite the expression.

$40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} = - ((1 + 8 x^{5}) (1 - 8 x^{5}))$

Simplify the right side.

Simplify $- ((1 + 8 x^{5}) (1 - 8 x^{5}))$ .

Expand $(1 + 8 x^{5}) (1 - 8 x^{5})$ using the FOIL Method.

Apply the distributive property.

$40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} = - (1 (1 - 8 x^{5}) + 8 x^{5} (1 - 8 x^{5}))$

Apply the distributive property.

$40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} = - (1 \cdot 1 + 1 (- 8 x^{5}) + 8 x^{5} (1 - 8 x^{5}))$

Apply the distributive property.

$40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} = - (1 \cdot 1 + 1 (- 8 x^{5}) + 8 x^{5} \cdot 1 + 8 x^{5} (- 8 x^{5}))$

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

$40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} = - (1 + 1 (- 8 x^{5}) + 8 x^{5} \cdot 1 + 8 x^{5} (- 8 x^{5}))$

Multiply $- 8 x^{5}$ by $1$ .

$40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} = - (1 - 8 x^{5} + 8 x^{5} \cdot 1 + 8 x^{5} (- 8 x^{5}))$

Multiply $8$ by $1$ .

$40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} = - (1 - 8 x^{5} + 8 x^{5} + 8 x^{5} (- 8 x^{5}))$

Rewrite using the commutative property of multiplication.

$40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} = - (1 - 8 x^{5} + 8 x^{5} + 8 \cdot - 8 x^{5} x^{5})$

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

Move $x^{5}$ .

$40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} = - (1 - 8 x^{5} + 8 x^{5} + 8 \cdot - 8 (x^{5} x^{5}))$

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

$40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} = - (1 - 8 x^{5} + 8 x^{5} + 8 \cdot - 8 x^{5 + 5})$

Add $5$ and $5$ .

$40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} = - (1 - 8 x^{5} + 8 x^{5} + 8 \cdot - 8 x^{10})$

Multiply $8$ by $- 8$ .

$40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} = - (1 - 8 x^{5} + 8 x^{5} - 64 x^{10})$

Add $- 8 x^{5}$ and $8 x^{5}$ .

$40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} = - (1 + 0 - 64 x^{10})$

Add $1$ and $0$ .

$40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} = - (1 - 64 x^{10})$

Apply the distributive property.

$40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} = - 1 \cdot 1 - (- 64 x^{10})$

Simplify the expression.

Multiply $- 1$ by $1$ .

$40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} = - 1 - (- 64 x^{10})$

Multiply $- 64$ by $- 1$ .

$40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} = - 1 + 64 x^{10}$

Reorder $- 1$ and $64 x^{10}$ .

$40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} = 64 x^{10} - 1$

Divide each term in $40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} = 64 x^{10} - 1$ by $40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}$ and simplify.

Divide each term in $40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} = 64 x^{10} - 1$ by $40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}$ .

$\frac{40 x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}} = \frac{64 x^{10}}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}} + \frac{- 1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

Simplify the left side.

Cancel the common factor of $40$ .

Cancel the common factor.

Rewrite the expression.

$\frac{x^{4} x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}} = \frac{64 x^{10}}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}} + \frac{- 1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

Cancel the common factor of $x^{4}$ .

Cancel the common factor.

Rewrite the expression.

$\frac{x' \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}} = \frac{64 x^{10}}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}} + \frac{- 1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

Cancel the common factor of $\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}$ .

Cancel the common factor.

Divide $x'$ by $1$ .

$x' = \frac{64 x^{10}}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}} + \frac{- 1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $64$ and $40$ .

Factor $8$ out of $64 x^{10}$ .

$x' = \frac{8 (8 x^{10})}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}} + \frac{- 1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

Cancel the common factors.

Factor $8$ out of $40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}$ .

$x' = \frac{8 (8 x^{10})}{8 (5 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})})} + \frac{- 1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

Cancel the common factor.

$x' = \frac{8 (8 x^{10})}{8 (5 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})})} + \frac{- 1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

Rewrite the expression.

$x' = \frac{8 x^{10}}{5 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}} + \frac{- 1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

Cancel the common factor of $x^{10}$ and $x^{4}$ .

Factor $x^{4}$ out of $8 x^{10}$ .

$x' = \frac{x^{4} (8 x^{6})}{5 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}} + \frac{- 1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

Cancel the common factors.

Factor $x^{4}$ out of $5 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}$ .

$x' = \frac{x^{4} (8 x^{6})}{x^{4} (5 \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})})} + \frac{- 1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

Cancel the common factor.

$x' = \frac{x^{4} (8 x^{6})}{x^{4} (5 \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})})} + \frac{- 1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

Rewrite the expression.

$x' = \frac{8 x^{6}}{5 \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}} + \frac{- 1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

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

$x' = \frac{8 x^{6}}{5 \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}} \cdot \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}} + \frac{- 1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

Combine and simplify the denominator.

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

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}} + \frac{- 1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

Move $\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}$ .

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})})} + \frac{- 1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

Raise $\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}$ to the power of $1$ .

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 ({\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}^{1} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})})} + \frac{- 1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

Raise $\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}$ to the power of $1$ .

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 ({\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}^{1} {\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}^{1})} + \frac{- 1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

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

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 {\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}^{1 + 1}} + \frac{- 1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

Add $1$ and $1$ .

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 {\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}^{2}} + \frac{- 1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

Rewrite ${\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}^{2}$ as $(1 + 8 x^{5}) (1 - 8 x^{5})$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}$ as ${((1 + 8 x^{5}) (1 - 8 x^{5}))}^{\frac{1}{2}}$ .

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 {({((1 + 8 x^{5}) (1 - 8 x^{5}))}^{\frac{1}{2}})}^{2}} + \frac{- 1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

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

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 {((1 + 8 x^{5}) (1 - 8 x^{5}))}^{\frac{1}{2} \cdot 2}} + \frac{- 1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

Combine $\frac{1}{2}$ and $2$ .

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 {((1 + 8 x^{5}) (1 - 8 x^{5}))}^{\frac{2}{2}}} + \frac{- 1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 {((1 + 8 x^{5}) (1 - 8 x^{5}))}^{\frac{2}{2}}} + \frac{- 1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

Rewrite the expression.

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 {((1 + 8 x^{5}) (1 - 8 x^{5}))}^{1}} + \frac{- 1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

Simplify.

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 ((1 + 8 x^{5}) (1 - 8 x^{5}))} + \frac{- 1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

Move the negative in front of the fraction.

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (1 + 8 x^{5}) (1 - 8 x^{5})} - \frac{1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

Multiply $\frac{1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$ by $\frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$ .

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (1 + 8 x^{5}) (1 - 8 x^{5})} - (\frac{1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}} \cdot \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}})$

Combine and simplify the denominator.

Multiply $\frac{1}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$ by $\frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$ .

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (1 + 8 x^{5}) (1 - 8 x^{5})} - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}$

Move $\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}$ .

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (1 + 8 x^{5}) (1 - 8 x^{5})} - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} (\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})})}$

Raise $\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}$ to the power of $1$ .

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (1 + 8 x^{5}) (1 - 8 x^{5})} - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} ({\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}^{1} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})})}$

Raise $\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}$ to the power of $1$ .

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (1 + 8 x^{5}) (1 - 8 x^{5})} - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} ({\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}^{1} {\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}^{1})}$

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

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (1 + 8 x^{5}) (1 - 8 x^{5})} - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} {\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}^{1 + 1}}$

Add $1$ and $1$ .

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (1 + 8 x^{5}) (1 - 8 x^{5})} - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} {\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}^{2}}$

Rewrite ${\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}^{2}$ as $(1 + 8 x^{5}) (1 - 8 x^{5})$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}$ as ${((1 + 8 x^{5}) (1 - 8 x^{5}))}^{\frac{1}{2}}$ .

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (1 + 8 x^{5}) (1 - 8 x^{5})} - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} {({((1 + 8 x^{5}) (1 - 8 x^{5}))}^{\frac{1}{2}})}^{2}}$

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

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (1 + 8 x^{5}) (1 - 8 x^{5})} - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} {((1 + 8 x^{5}) (1 - 8 x^{5}))}^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (1 + 8 x^{5}) (1 - 8 x^{5})} - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} {((1 + 8 x^{5}) (1 - 8 x^{5}))}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (1 + 8 x^{5}) (1 - 8 x^{5})} - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} {((1 + 8 x^{5}) (1 - 8 x^{5}))}^{\frac{2}{2}}}$

Rewrite the expression.

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (1 + 8 x^{5}) (1 - 8 x^{5})} - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} {((1 + 8 x^{5}) (1 - 8 x^{5}))}^{1}}$

Simplify.

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (1 + 8 x^{5}) (1 - 8 x^{5})} - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} ((1 + 8 x^{5}) (1 - 8 x^{5}))}$

Simplify the denominator.

Simplify the left side.

Move $\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}$ .

Raise $\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}$ to the power of $1$ .

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

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (1 + 8 x^{5}) (1 - 8 x^{5})} - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} {\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}^{1 + 1}}$

Add $1$ and $1$ .

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (1 + 8 x^{5}) (1 - 8 x^{5})} - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} {\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}^{2}}$

Rewrite ${\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}^{2}$ as $(1 + 8 x^{5}) (1 - 8 x^{5})$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}$ as ${((1 + 8 x^{5}) (1 - 8 x^{5}))}^{\frac{1}{2}}$ .

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (1 + 8 x^{5}) (1 - 8 x^{5})} - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} {({((1 + 8 x^{5}) (1 - 8 x^{5}))}^{\frac{1}{2}})}^{2}}$

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

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (1 + 8 x^{5}) (1 - 8 x^{5})} - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} {((1 + 8 x^{5}) (1 - 8 x^{5}))}^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (1 + 8 x^{5}) (1 - 8 x^{5})} - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} {((1 + 8 x^{5}) (1 - 8 x^{5}))}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (1 + 8 x^{5}) (1 - 8 x^{5})} - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} {((1 + 8 x^{5}) (1 - 8 x^{5}))}^{\frac{2}{2}}}$

Rewrite the expression.

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (1 + 8 x^{5}) (1 - 8 x^{5})} - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} {((1 + 8 x^{5}) (1 - 8 x^{5}))}^{1}}$

Simplify.

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (1 + 8 x^{5}) (1 - 8 x^{5})} - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} ((1 + 8 x^{5}) (1 - 8 x^{5}))}$

Remove unnecessary parentheses.

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (1 + 8 x^{5}) (1 - 8 x^{5})} - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} (1 + 8 x^{5}) (1 - 8 x^{5})}$

To write $\frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (1 + 8 x^{5}) (1 - 8 x^{5})}$ as a fraction with a common denominator, multiply by $\frac{8 x^{4}}{8 x^{4}}$ .

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (1 + 8 x^{5}) (1 - 8 x^{5})} \cdot \frac{8 x^{4}}{8 x^{4}} - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} (1 + 8 x^{5}) (1 - 8 x^{5})}$

Write each expression with a common denominator of $40 (1 + 8 x^{5}) (1 - 8 x^{5}) x^{4}$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{5 (1 + 8 x^{5}) (1 - 8 x^{5})}$ by $\frac{8 x^{4}}{8 x^{4}}$ .

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} (8 x^{4})}{5 (1 + 8 x^{5}) (1 - 8 x^{5}) (8 x^{4})} - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} (1 + 8 x^{5}) (1 - 8 x^{5})}$

Multiply $8$ by $5$ .

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} (8 x^{4})}{40 (1 + 8 x^{5}) (1 - 8 x^{5}) x^{4}} - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} (1 + 8 x^{5}) (1 - 8 x^{5})}$

Reorder the factors of $40 (1 + 8 x^{5}) (1 - 8 x^{5}) x^{4}$ .

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} (8 x^{4})}{40 x^{4} (1 + 8 x^{5}) (1 - 8 x^{5})} - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} (1 + 8 x^{5}) (1 - 8 x^{5})}$

Combine the numerators over the common denominator.

$x' = \frac{8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} (8 x^{4}) - \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} (1 + 8 x^{5}) (1 - 8 x^{5})}$

Simplify the numerator.

Factor $\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}$ out of $8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} (8 x^{4}) - \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}$ .

Factor $\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}$ out of $8 x^{6} \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} (8 x^{4})$ .

$x' = \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} (8 x^{6} (8 x^{4})) - \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4} (1 + 8 x^{5}) (1 - 8 x^{5})}$

Factor $\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}$ out of $- \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}$ .

$x' = \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} (8 x^{6} (8 x^{4})) + \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} \cdot - 1}{40 x^{4} (1 + 8 x^{5}) (1 - 8 x^{5})}$

Factor $\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}$ out of $\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} (8 x^{6} (8 x^{4})) + \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} \cdot - 1$ .

$x' = \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} (8 x^{6} (8 x^{4}) - 1)}{40 x^{4} (1 + 8 x^{5}) (1 - 8 x^{5})}$

Multiply $8$ by $8$ .

$x' = \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} (64 x^{6} x^{4} - 1)}{40 x^{4} (1 + 8 x^{5}) (1 - 8 x^{5})}$

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

Move $x^{4}$ .

$x' = \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} (64 (x^{4} x^{6}) - 1)}{40 x^{4} (1 + 8 x^{5}) (1 - 8 x^{5})}$

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

$x' = \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} (64 x^{4 + 6} - 1)}{40 x^{4} (1 + 8 x^{5}) (1 - 8 x^{5})}$

Add $4$ and $6$ .

$x' = \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} (64 x^{10} - 1)}{40 x^{4} (1 + 8 x^{5}) (1 - 8 x^{5})}$

Rewrite $64 x^{10}$ as ${(8 x^{5})}^{2}$ .

$x' = \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} ({(8 x^{5})}^{2} - 1)}{40 x^{4} (1 + 8 x^{5}) (1 - 8 x^{5})}$

Rewrite $1$ as $1^{2}$ .

$x' = \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} ({(8 x^{5})}^{2} - 1^{2})}{40 x^{4} (1 + 8 x^{5}) (1 - 8 x^{5})}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 8 x^{5}$ and $b = 1$ .

$x' = \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} (8 x^{5} + 1) (8 x^{5} - 1)}{40 x^{4} (1 + 8 x^{5}) (1 - 8 x^{5})}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $8 x^{5} + 1$ and $1 + 8 x^{5}$ .

Reorder terms.

$x' = \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} (8 x^{5} + 1) (8 x^{5} - 1)}{40 x^{4} (8 x^{5} + 1) (1 - 8 x^{5})}$

Cancel the common factor.

$x' = \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} (8 x^{5} + 1) (8 x^{5} - 1)}{40 x^{4} (8 x^{5} + 1) (1 - 8 x^{5})}$

Rewrite the expression.

$x' = \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} (8 x^{5} - 1)}{40 x^{4} (1 - 8 x^{5})}$

Cancel the common factor of $8 x^{5} - 1$ and $1 - 8 x^{5}$ .

Factor $- 1$ out of $8 x^{5}$ .

$x' = \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} (- (- 8 x^{5}) - 1)}{40 x^{4} (1 - 8 x^{5})}$

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

$x' = \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} (- (- 8 x^{5}) - 1 (1))}{40 x^{4} (1 - 8 x^{5})}$

Factor $- 1$ out of $- (- 8 x^{5}) - 1 (1)$ .

$x' = \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} (- (- 8 x^{5} + 1))}{40 x^{4} (1 - 8 x^{5})}$

Rewrite $- (- 8 x^{5} + 1)$ as $- 1 (- 8 x^{5} + 1)$ .

$x' = \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} (- 1 (- 8 x^{5} + 1))}{40 x^{4} (1 - 8 x^{5})}$

Reorder terms.

$x' = \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} (- 1 (- 8 x^{5} + 1))}{40 x^{4} (- 8 x^{5} + 1)}$

Cancel the common factor.

$x' = \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} (- 1 (- 8 x^{5} + 1))}{40 x^{4} (- 8 x^{5} + 1)}$

Rewrite the expression.

$x' = \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})} \cdot (- 1)}{40 x^{4}}$

Simplify the expression.

Move $- 1$ to the left of $\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}$ .

$x' = \frac{- 1 \cdot \sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4}}$

Move the negative in front of the fraction.

$x' = - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4}}$

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

$\frac{d x}{d y} = - \frac{\sqrt{(1 + 8 x^{5}) (1 - 8 x^{5})}}{40 x^{4}}$

Find dx/dy y=arccos(8x^5)

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