Calculus Examples

$y = \arccos (x^{2})$

Step 1

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

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Step 1.1

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

$\frac{d}{d u} [\arccos (u)] \frac{d}{d x} [x^{2}]$

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 2

Differentiate using the Power Rule.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Multiply $2$ by $2$ .

$- \frac{1}{\sqrt{1 - x^{4}}} \frac{d}{d x} [x^{2}]$

Step 2.2

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

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

Step 2.3

Combine fractions.

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Step 2.3.1

Multiply $2$ by $- 1$ .

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

Move the negative in front of the fraction.

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