Problem 39 - Calculus of a Single Variable

Question

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

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = 2\arcsin(x - 1)\) is \(f'(x) = \frac{2}{\sqrt{2x - x^2}}\).

Step 1: Identify the outer function and the inner function

In the given function \(f(x) = 2\arcsin(x-1)\), the outer function is \(2\arcsin(u)\) and the inner function is \(u = x - 1\). The chain rule states that the derivative of a composite function is the derivative of the outer function, evaluated at the inner function, times the derivative of the inner function.

Step 2: Find the derivatives of the outer function and the inner function

The derivative of the outer function \(2\arcsin(u)\) with respect to \(u\) is \(\frac{2}{\sqrt{1-u^2}}\). The derivative of the inner function \(x - 1\) with respect to \(x\) is 1.

Step 3: Apply the chain rule

The chain rule states that the derivative of the composite function \(f(x)\) with respect to \(x\) is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Therefore, \(f'(x) = \frac{2}{\sqrt{1-(x - 1)^2}} \cdot 1\).

Step 4: Simplify the derivative

The expression inside the square root \(1-(x - 1)^2 = 2x - x^2\). Consequently, the derivative simplifies to \(f'(x) = \frac{2}{\sqrt{2x - x^2}}\).