Calculus Examples

$y = arcsec (\frac{1}{9 t^{4}})$

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 $g (t) = \frac{1}{9 t^{4}}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{1}{9 t^{4}}$ .

$\frac{d}{d u} [arcsec (u)] \frac{d}{d t} [\frac{1}{9 t^{4}}]$

Step 1.2

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

$\frac{1}{u \sqrt{u^{2} - 1}} \frac{d}{d t} [\frac{1}{9 t^{4}}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{1}{9 t^{4}}$ .

$\frac{1}{\frac{1}{9 t^{4}} \sqrt{{(\frac{1}{9 t^{4}})}^{2} - 1}} \frac{d}{d t} [\frac{1}{9 t^{4}}]$

Step 2

Combine $\frac{1}{9 t^{4}}$ and $\sqrt{{(\frac{1}{9 t^{4}})}^{2} - 1}$ .

$\frac{1}{\frac{\sqrt{{(\frac{1}{9 t^{4}})}^{2} - 1}}{9 t^{4}}} \frac{d}{d t} [\frac{1}{9 t^{4}}]$

Step 3

Multiply by the reciprocal of the fraction to divide by $\frac{\sqrt{{(\frac{1}{9 t^{4}})}^{2} - 1}}{9 t^{4}}$ .

$1 \frac{9 t^{4}}{\sqrt{{(\frac{1}{9 t^{4}})}^{2} - 1}} \frac{d}{d t} [\frac{1}{9 t^{4}}]$

Step 4

Multiply $\frac{9 t^{4}}{\sqrt{{(\frac{1}{9 t^{4}})}^{2} - 1}}$ by $1$ .

$\frac{9 t^{4}}{\sqrt{{(\frac{1}{9 t^{4}})}^{2} - 1}} \frac{d}{d t} [\frac{1}{9 t^{4}}]$

Step 5

Since $\frac{1}{9}$ is constant with respect to $t$ , the derivative of $\frac{1}{9 t^{4}}$ with respect to $t$ is $\frac{1}{9} \frac{d}{d t} [\frac{1}{t^{4}}]$ .

$\frac{9 t^{4}}{\sqrt{{(\frac{1}{9 t^{4}})}^{2} - 1}} (\frac{1}{9} \frac{d}{d t} [\frac{1}{t^{4}}])$

Step 6

Simplify terms.

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Step 6.1

Multiply $\frac{1}{9}$ by $\frac{9 t^{4}}{\sqrt{{(\frac{1}{9 t^{4}})}^{2} - 1}}$ .

$\frac{9 t^{4}}{9 \sqrt{{(\frac{1}{9 t^{4}})}^{2} - 1}} \frac{d}{d t} [\frac{1}{t^{4}}]$

Step 6.2

Cancel the common factor of $9$ .

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Step 6.2.1

Cancel the common factor.

$\frac{9 t^{4}}{9 \sqrt{{(\frac{1}{9 t^{4}})}^{2} - 1}} \frac{d}{d t} [\frac{1}{t^{4}}]$

Step 6.2.2

Rewrite the expression.

$\frac{t^{4}}{\sqrt{{(\frac{1}{9 t^{4}})}^{2} - 1}} \frac{d}{d t} [\frac{1}{t^{4}}]$

Step 6.3

Apply basic rules of exponents.

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Step 6.3.1

Rewrite $\frac{1}{t^{4}}$ as ${(t^{4})}^{- 1}$ .

$\frac{t^{4}}{\sqrt{{(\frac{1}{9 t^{4}})}^{2} - 1}} \frac{d}{d t} [{(t^{4})}^{- 1}]$

Step 6.3.2

Multiply the exponents in ${(t^{4})}^{- 1}$ .

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Step 6.3.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{t^{4}}{\sqrt{{(\frac{1}{9 t^{4}})}^{2} - 1}} \frac{d}{d t} [t^{4 \cdot - 1}]$

Step 6.3.2.2

Multiply $4$ by $- 1$ .

$\frac{t^{4}}{\sqrt{{(\frac{1}{9 t^{4}})}^{2} - 1}} \frac{d}{d t} [t^{- 4}]$

Step 7

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = - 4$ .

$\frac{t^{4}}{\sqrt{{(\frac{1}{9 t^{4}})}^{2} - 1}} (- 4 t^{- 5})$

Step 8

Combine fractions.

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Step 8.1

Combine $- 4$ and $\frac{t^{4}}{\sqrt{{(\frac{1}{9 t^{4}})}^{2} - 1}}$ .

$\frac{- 4 t^{4}}{\sqrt{{(\frac{1}{9 t^{4}})}^{2} - 1}} t^{- 5}$

Step 8.2

Combine $\frac{- 4 t^{4}}{\sqrt{{(\frac{1}{9 t^{4}})}^{2} - 1}}$ and $t^{- 5}$ .

$\frac{- 4 t^{4} t^{- 5}}{\sqrt{{(\frac{1}{9 t^{4}})}^{2} - 1}}$

Step 9

Multiply $t^{4}$ by $t^{- 5}$ by adding the exponents.

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Step 9.1

Move $t^{- 5}$ .

$\frac{- 4 (t^{- 5} t^{4})}{\sqrt{{(\frac{1}{9 t^{4}})}^{2} - 1}}$

Step 9.2

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

$\frac{- 4 t^{- 5 + 4}}{\sqrt{{(\frac{1}{9 t^{4}})}^{2} - 1}}$

Step 9.3

Add $- 5$ and $4$ .

$\frac{- 4 t^{- 1}}{\sqrt{{(\frac{1}{9 t^{4}})}^{2} - 1}}$

Step 10

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

$\frac{- 4}{\sqrt{{(\frac{1}{9 t^{4}})}^{2} - 1} t}$

Step 11

Move the negative in front of the fraction.

$- \frac{4}{\sqrt{{(\frac{1}{9 t^{4}})}^{2} - 1} t}$

Step 12

Simplify.

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Step 12.1

Apply the product rule to $\frac{1}{9 t^{4}}$ .

$- \frac{4}{\sqrt{\frac{1^{2}}{{(9 t^{4})}^{2}} - 1} t}$

Step 12.2

Apply the product rule to $9 t^{4}$ .

$- \frac{4}{\sqrt{\frac{1^{2}}{9^{2} {(t^{4})}^{2}} - 1} t}$

Step 12.3

Combine terms.

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Step 12.3.1

One to any power is one.

$- \frac{4}{\sqrt{\frac{1}{9^{2} {(t^{4})}^{2}} - 1} t}$

Step 12.3.2

Raise $9$ to the power of $2$ .

$- \frac{4}{\sqrt{\frac{1}{81 {(t^{4})}^{2}} - 1} t}$

Step 12.3.3

Multiply the exponents in ${(t^{4})}^{2}$ .

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Step 12.3.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{4}{\sqrt{\frac{1}{81 t^{4 \cdot 2}} - 1} t}$

Step 12.3.3.2

Multiply $4$ by $2$ .

$- \frac{4}{\sqrt{\frac{1}{81 t^{8}} - 1} t}$