Calculus Examples

$y = arccsc (e^{4 t})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (y) = \frac{d}{d t} (arccsc (e^{4 t}))$

Step 2

The derivative of $y$ with respect to $t$ 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 $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = arccsc (t)$ and $g (t) = e^{4 t}$ .

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Step 3.1.1

To apply the Chain Rule, set $u_{1}$ as $e^{4 t}$ .

$\frac{d}{d u_{1}} [arccsc (u_{1})] \frac{d}{d t} [e^{4 t}]$

Step 3.1.2

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

$- \frac{1}{u_{1} \sqrt{{u_{1}}^{2} - 1}} \frac{d}{d t} [e^{4 t}]$

Step 3.1.3

Replace all occurrences of $u_{1}$ with $e^{4 t}$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Multiply $2$ by $4$ .

$- \frac{1}{e^{4 t} \sqrt{e^{8 t} - 1}} \frac{d}{d t} [e^{4 t}]$

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = e^{t}$ and $g (t) = 4 t$ .

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Step 3.3.1

To apply the Chain Rule, set $u_{2}$ as $4 t$ .

$- \frac{1}{e^{4 t} \sqrt{e^{8 t} - 1}} (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d t} [4 t])$

Step 3.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$- \frac{1}{e^{4 t} \sqrt{e^{8 t} - 1}} (e^{u_{2}} \frac{d}{d t} [4 t])$

Step 3.3.3

Replace all occurrences of $u_{2}$ with $4 t$ .

$- \frac{1}{e^{4 t} \sqrt{e^{8 t} - 1}} (e^{4 t} \frac{d}{d t} [4 t])$

Step 3.4

Differentiate.

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Step 3.4.1

Combine $e^{4 t}$ and $\frac{1}{e^{4 t} \sqrt{e^{8 t} - 1}}$ .

$- \frac{e^{4 t}}{e^{4 t} \sqrt{e^{8 t} - 1}} \frac{d}{d t} [4 t]$

Step 3.4.2

Cancel the common factor of $e^{4 t}$ .

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Step 3.4.2.1

Cancel the common factor.

$- \frac{e^{4 t}}{e^{4 t} \sqrt{e^{8 t} - 1}} \frac{d}{d t} [4 t]$

Step 3.4.2.2

Rewrite the expression.

$- \frac{1}{\sqrt{e^{8 t} - 1}} \frac{d}{d t} [4 t]$

Step 3.4.3

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

$- \frac{1}{\sqrt{e^{8 t} - 1}} (4 \frac{d}{d t} [t])$

Step 3.4.4

Combine fractions.

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Step 3.4.4.1

Multiply $4$ by $- 1$ .

$- 4 \frac{1}{\sqrt{e^{8 t} - 1}} \frac{d}{d t} [t]$

Step 3.4.4.2

Combine $- 4$ and $\frac{1}{\sqrt{e^{8 t} - 1}}$ .

$\frac{- 4}{\sqrt{e^{8 t} - 1}} \frac{d}{d t} [t]$

Step 3.4.4.3

Move the negative in front of the fraction.

$- \frac{4}{\sqrt{e^{8 t} - 1}} \frac{d}{d t} [t]$

Step 3.4.5

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

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

Step 3.4.6

Multiply $- 1$ by $1$ .

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

Step 4

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

$y' = - \frac{4}{\sqrt{e^{8 t} - 1}}$

Step 5

Replace $y'$ with $\frac{d y}{d t}$ .

$\frac{d y}{d t} = - \frac{4}{\sqrt{e^{8 t} - 1}}$