Calculus Examples

$y = - \arccos (\frac{1}{x^{3}})$

Step 1

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

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

Step 2

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 $g (x) = \frac{1}{x^{3}}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $\frac{1}{x^{3}}$ .

$- (\frac{d}{d u} [\arccos (u)] \frac{d}{d x} [\frac{1}{x^{3}}])$

Step 2.2

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

$- (- \frac{1}{\sqrt{1 - u^{2}}} \frac{d}{d x} [\frac{1}{x^{3}}])$

Step 2.3

Replace all occurrences of $u$ with $\frac{1}{x^{3}}$ .

$- (- \frac{1}{\sqrt{1 - {(\frac{1}{x^{3}})}^{2}}} \frac{d}{d x} [\frac{1}{x^{3}}])$

Step 3

Differentiate using the Power Rule.

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Step 3.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{\sqrt{1 - {(\frac{1}{x^{3}})}^{2}}} \frac{d}{d x} [\frac{1}{x^{3}}])$

Step 3.2

Simplify the expression.

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Step 3.2.1

Multiply $\frac{1}{\sqrt{1 - {(\frac{1}{x^{3}})}^{2}}}$ by $1$ .

$\frac{1}{\sqrt{1 - {(\frac{1}{x^{3}})}^{2}}} \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 3.2.2

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

$\frac{1}{\sqrt{1 - {(\frac{1}{x^{3}})}^{2}}} \frac{d}{d x} [{(x^{3})}^{- 1}]$

Step 3.2.3

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 3.2.3.1

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

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

Step 3.2.3.2

Multiply $3$ by $- 1$ .

$\frac{1}{\sqrt{1 - {(\frac{1}{x^{3}})}^{2}}} \frac{d}{d x} [x^{- 3}]$

Step 3.3

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

$\frac{1}{\sqrt{1 - {(\frac{1}{x^{3}})}^{2}}} (- 3 x^{- 4})$

Step 3.4

Combine fractions.

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Step 3.4.1

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

$\frac{- 3}{\sqrt{1 - {(\frac{1}{x^{3}})}^{2}}} x^{- 4}$

Step 3.4.2

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

$\frac{- 3 x^{- 4}}{\sqrt{1 - {(\frac{1}{x^{3}})}^{2}}}$

Step 3.4.3

Simplify the expression.

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Step 3.4.3.1

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

$\frac{- 3}{\sqrt{1 - {(\frac{1}{x^{3}})}^{2}} x^{4}}$

Step 3.4.3.2

Move the negative in front of the fraction.

$- \frac{3}{\sqrt{1 - {(\frac{1}{x^{3}})}^{2}} x^{4}}$

Step 4

Simplify.

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Step 4.1

Apply the product rule to $\frac{1}{x^{3}}$ .

$- \frac{3}{\sqrt{1 - \frac{1^{2}}{{(x^{3})}^{2}}} x^{4}}$

Step 4.2

Combine terms.

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Step 4.2.1

One to any power is one.

$- \frac{3}{\sqrt{1 - \frac{1}{{(x^{3})}^{2}}} x^{4}}$

Step 4.2.2

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 4.2.2.1

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

$- \frac{3}{\sqrt{1 - \frac{1}{x^{3 \cdot 2}}} x^{4}}$

Step 4.2.2.2

Multiply $3$ by $2$ .

$- \frac{3}{\sqrt{1 - \frac{1}{x^{6}}} x^{4}}$