Question

Find the derivative of f(x)=2arcsin(x-1)

Answer 1

Since
`(\mathrm d)/(\mathrm dx)\arcsin x=\frac1{√(1-x^2)}`, you have

`\displaystyle(\mathrm d)/(\mathrm dx)2\arcsin(x-1)=2(\mathrm d)/(\mathrm dx)\arcsin(x-1)=2((\mathrm d)/(\mathrm dx)(x-1))/(√(1-(x-1)^2))=\frac2{√(2x-x^2)}`

Answer 2

`\displaystyle f'(x) = (2)/(√(1 - (x - 1)^2))`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:
`\displaystyle (d)/(dx) [cf(x)] = c \cdot f'(x)`

Derivative Property [Addition/Subtraction]:
`\displaystyle (d)/(dx)[f(x) + g(x)] = (d)/(dx)[f(x)] + (d)/(dx)[g(x)]`

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Derivative Rule [Chain Rule]:
`\displaystyle (d)/(dx)[f(g(x))] =f'(g(x)) \cdot g'(x)`

Explanation:

Step 1: Define

Identify

`\displaystyle f(x) = 2 \arcsin (x - 1)`

Step 2: Differentiate

1. Derivative Property [Multiplied Constant]:
   `\displaystyle f'(x) = 2 (d)/(dx)[\arcsin (x - 1)]`
2. Trigonometric Differentiation [Derivative Rule - Chain Rule]:
   `\displaystyle f'(x) = 2 \bigg( ((x - 1)')/(√(1 - (x - 1)^2)) \bigg)`
3. Basic Power Rule [Derivative Properties]:
   `\displaystyle f'(x) = 2 \bigg( (1)/(√(1 - (x - 1)^2)) \bigg)`
4. Simplify:
   `\displaystyle f'(x) = (2)/(√(1 - (x - 1)^2))`

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation