Calculus Examples

$2 \arcsin (x - 1)$

Step 1

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

$2 \frac{d}{d x} [\arcsin (x - 1)]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \arcsin (x)$ and $g (x) = x - 1$ .

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Step 2.1

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

$2 (\frac{d}{d u} [\arcsin (u)] \frac{d}{d x} [x - 1])$

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Simplify the expression.

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Step 3.5.1

Add $1$ and $0$ .

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

Step 3.5.2

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

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