Calculus Examples

$y = arcsec (\frac{1}{t})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = arcsec (t)$ and $g (t) = \frac{1}{t}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{1}{t}$ .

$\frac{d}{d u} [arcsec (u)] \frac{d}{d t} [\frac{1}{t}]$

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $\frac{1}{t}$ .

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

Step 2

Combine $\frac{1}{t}$ and $\sqrt{{(\frac{1}{t})}^{2} - 1}$ .

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

Step 3

Multiply by the reciprocal of the fraction to divide by $\frac{\sqrt{{(\frac{1}{t})}^{2} - 1}}{t}$ .

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

Step 4

Simplify the expression.

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Step 4.1

Multiply $\frac{t}{\sqrt{{(\frac{1}{t})}^{2} - 1}}$ by $1$ .

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

Step 4.2

Rewrite $\frac{1}{t}$ as $t^{- 1}$ .

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

Step 5

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

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

Step 6

Combine $\frac{t}{\sqrt{{(\frac{1}{t})}^{2} - 1}}$ and $t^{- 2}$ .

$- \frac{t \cdot t^{- 2}}{\sqrt{{(\frac{1}{t})}^{2} - 1}}$

Step 7

Multiply $t$ by $t^{- 2}$ by adding the exponents.

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Step 7.1

Multiply $t$ by $t^{- 2}$ .

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Step 7.1.1

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

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

Step 7.1.2

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

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Move $t^{- 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 9

Simplify.

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Step 9.1

Apply the product rule to $\frac{1}{t}$ .

$- \frac{1}{\sqrt{\frac{1^{2}}{t^{2}} - 1} t}$

Step 9.2

One to any power is one.

$- \frac{1}{\sqrt{\frac{1}{t^{2}} - 1} t}$