Calculus Examples

$y = \arcsin (\cos (x))$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u$ as $\cos (x)$ .

$\frac{d}{d u} [\arcsin (u)] \frac{d}{d x} [\cos (x)]$

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $\cos (x)$ .

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

Step 2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Apply pythagorean identity.

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

Step 4.2

Pull terms out from under the radical, assuming positive real numbers.

$- \frac{\sin (x)}{\sin (x)}$

Step 4.3

Cancel the common factor of $\sin (x)$ .

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Step 4.3.1

Cancel the common factor.

$- \frac{\sin (x)}{\sin (x)}$

Step 4.3.2

Rewrite the expression.

$- 1 \cdot 1$

Step 4.4

Multiply $- 1$ by $1$ .

$- 1$