Calculus Examples

$y = arcsec (\cos (2 x))$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (arcsec (\cos (2 x)))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $g (x) = \cos (2 x)$ .

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Step 3.1.1

To apply the Chain Rule, set $u_{1}$ as $\cos (2 x)$ .

$\frac{d}{d u_{1}} [arcsec (u_{1})] \frac{d}{d x} [\cos (2 x)]$

Step 3.1.2

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

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

Step 3.1.3

Replace all occurrences of $u_{1}$ with $\cos (2 x)$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

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

Step 3.2.3

Replace all occurrences of $u_{2}$ with $2 x$ .

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

Step 3.3

Differentiate.

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Step 3.3.1

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

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

Step 3.3.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 3.3.3

Combine fractions.

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Step 3.3.3.1

Multiply $2$ by $- 1$ .

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

Step 3.3.3.2

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

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

Step 3.3.3.3

Move the negative in front of the fraction.

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

Step 3.3.4

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

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

Step 3.3.5

Multiply $- 1$ by $1$ .

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

Step 3.4

Simplify.

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.4.4

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

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

Step 3.4.5

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

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

Step 3.4.6

Apply pythagorean identity.

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

Step 3.4.7

Simplify the denominator.

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Step 3.4.7.1

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

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

Step 3.4.7.2

Pull terms out from under the radical.

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

Step 3.4.7.3

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

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

Step 3.4.8

Cancel the common factor of $\sin (2 x)$ .

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Step 3.4.8.1

Cancel the common factor.

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

Step 3.4.8.2

Rewrite the expression.

$- \frac{2}{\cos (2 x) i}$

Step 3.4.9

Separate fractions.

$- (\frac{2}{i} \cdot \frac{1}{\cos (2 x)})$

Step 3.4.10

Convert from $\frac{1}{\cos (2 x)}$ to $\sec (2 x)$ .

$- (\frac{2}{i} \sec (2 x))$

Step 3.4.11

Multiply the numerator and denominator of $\frac{2}{i}$ by the conjugate of $i$ to make the denominator real.

$- (\frac{2}{i} \cdot \frac{i}{i} \sec (2 x))$

Step 3.4.12

Multiply.

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Step 3.4.12.1

Combine.

$- (\frac{2 i}{i i} \sec (2 x))$

Step 3.4.12.2

Simplify the denominator.

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Step 3.4.12.2.1

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

$- (\frac{2 i}{i^{1} i} \sec (2 x))$

Step 3.4.12.2.2

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

$- (\frac{2 i}{i^{1} i^{1}} \sec (2 x))$

Step 3.4.12.2.3

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

$- (\frac{2 i}{i^{1 + 1}} \sec (2 x))$

Step 3.4.12.2.4

Add $1$ and $1$ .

$- (\frac{2 i}{i^{2}} \sec (2 x))$

Step 3.4.12.2.5

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

$- (\frac{2 i}{- 1} \sec (2 x))$

Step 3.4.13

Move the negative one from the denominator of $\frac{2 i}{- 1}$ .

$- (- 1 \cdot (2 i) \sec (2 x))$

Step 3.4.14

Rewrite $- 1 \cdot (2 i)$ as $- (2 i)$ .

$- (- (2 i) \sec (2 x))$

Step 3.4.15

Multiply $2$ by $- 1$ .

$- (- 2 i \sec (2 x))$

Step 3.4.16

Multiply $- 2$ by $- 1$ .

$2 i \sec (2 x)$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 2 i \sec (2 x)$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 2 i \sec (2 x)$