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

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

Differentiate both sides of the equation.

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

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

$y'$

Differentiate the right side of the equation.

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

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

$\frac{d}{d u} [\arccos (u)] \frac{d}{d x} [8 x^{5}]$

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

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

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

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

Differentiate.

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

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

Simplify the expression.

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

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

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

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

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

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

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

Multiply $5$ by $2$ .

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

Multiply $64$ by $- 1$ .

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

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

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

Combine fractions.

Multiply $8$ by $- 1$ .

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

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

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

Move the negative in front of the fraction.

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

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

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

Combine fractions.

Multiply $5$ by $- 1$ .

$- 5 \frac{8}{\sqrt{1 - 64 x^{10}}} x^{4}$

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

$\frac{- 5 \cdot 8}{\sqrt{1 - 64 x^{10}}} x^{4}$

Multiply $- 5$ by $8$ .

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

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

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

Move the negative in front of the fraction.

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

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

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

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

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

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

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