Calculus Examples

$\frac{d}{d x} (arcsec (\sin (x)))$

Step 1

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

Tap for more steps...

Step 1.1

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

$\frac{d}{d u} [arcsec (u)] \frac{d}{d x} [\sin (x)]$

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

Reorder terms.

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

Step 4.2

Reorder $\sin^{2} (x)$ and $- 1$ .

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

Step 4.3

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 4.4

Factor $- 1$ out of $\sin^{2} (x)$ .

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

Step 4.5

Factor $- 1$ out of $- 1 (1) - 1 (- \sin^{2} (x))$ .

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

Step 4.6

Rewrite $- 1 (1 - \sin^{2} (x))$ as $- (1 - \sin^{2} (x))$ .

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

Step 4.7

Apply pythagorean identity.

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

Step 4.8

Simplify the denominator.

Tap for more steps...

Step 4.8.1

Reorder $- 1$ and $\cos^{2} (x)$ .

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

Step 4.8.2

Pull terms out from under the radical.

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

Step 4.8.3

Rewrite $\sqrt{- 1}$ as $i$ .

$\frac{\cos (x)}{\cos (x) i \sin (x)}$

Step 4.9

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

Tap for more steps...

Step 4.9.1

Cancel the common factor.

$\frac{\cos (x)}{\cos (x) i \sin (x)}$

Step 4.9.2

Rewrite the expression.

$\frac{1}{i \sin (x)}$

Step 4.10

Separate fractions.

$\frac{1}{i} \cdot \frac{1}{\sin (x)}$

Step 4.11

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\frac{1}{i} \csc (x)$

Step 4.12

Multiply the numerator and denominator of $\frac{1}{i}$ by the conjugate of $i$ to make the denominator real.

$\frac{1}{i} \cdot \frac{i}{i} \csc (x)$

Step 4.13

Multiply.

Tap for more steps...

Step 4.13.1

Combine.

$\frac{1 i}{i i} \csc (x)$

Step 4.13.2

Multiply $i$ by $1$ .

$\frac{i}{i i} \csc (x)$

Step 4.13.3

Simplify the denominator.

Tap for more steps...

Step 4.13.3.1

Raise $i$ to the power of $1$ .

$\frac{i}{i^{1} i} \csc (x)$

Step 4.13.3.2

Raise $i$ to the power of $1$ .

$\frac{i}{i^{1} i^{1}} \csc (x)$

Step 4.13.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{i}{i^{1 + 1}} \csc (x)$

Step 4.13.3.4

Add $1$ and $1$ .

$\frac{i}{i^{2}} \csc (x)$

Step 4.13.3.5

Rewrite $i^{2}$ as $- 1$ .

$\frac{i}{- 1} \csc (x)$

Step 4.14

Move the negative one from the denominator of $\frac{i}{- 1}$ .

$- 1 \cdot i \csc (x)$

Step 4.15

Rewrite $- 1 \cdot i$ as $- i$ .

$- i \csc (x)$