Calculus Examples

$s (t) = 2 \arccos (t - 1)$

Step 1

Since $2$ is constant with respect to $t$ , the derivative of $2 \arccos (t - 1)$ with respect to $t$ is $2 \frac{d}{d t} [\arccos (t - 1)]$ .

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

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u$ as $t - 1$ .

$2 (\frac{d}{d u} [\arccos (u)] \frac{d}{d t} [t - 1])$

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $t - 1$ .

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

Step 3

Differentiate.

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Step 3.1

Multiply $- 1$ by $2$ .

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

Step 3.2

Combine fractions.

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Step 3.2.1

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

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

Step 3.2.2

Move the negative in front of the fraction.

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

Step 3.3

By the Sum Rule, the derivative of $t - 1$ with respect to $t$ is $\frac{d}{d t} [t] + \frac{d}{d t} [- 1]$ .

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

Step 3.4

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

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

Step 3.5

Since $- 1$ is constant with respect to $t$ , the derivative of $- 1$ with respect to $t$ is $0$ .

$- \frac{2}{\sqrt{1 - {(t - 1)}^{2}}} (1 + 0)$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $1$ and $0$ .

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

Step 3.6.2

Multiply $- 1$ by $1$ .

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