Calculus Examples

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

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos^{-1} (x)$ and $g (x) = \sin (x)$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

Apply pythagorean identity.

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

Step 3.3

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

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

Step 3.4

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

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Rewrite the expression.

$- 1 \cdot 1$

Step 3.5

Multiply $- 1$ by $1$ .

$- 1$