Problem 53 - Calculus of a Single Variable

Question

Find the derivative of the function. \(y=x \arcsin x+\sqrt{1-x^{2}}\)

Step-by-Step Solution

Answer

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

Step 1: Apply the Product Rule

The derivative of the first term, \(x \arcsin x\), can be found by applying the product rule which states that the derivative of two multiplied functions is the first function times the derivative of the second plus the second function times the derivative of the first. In this case, the first function is \(x\) and the second function is \(\arcsin x\). The derivative of \(x\) is \(1\) and the derivative of \(\arcsin x\) is \(1 / \sqrt{1 - x^2}\). Thus, the derivative of the first term is: \(x \cdot (1 / \sqrt{1 - x^2}) + \arcsin x \cdot 1 = \frac{x}{\sqrt{1 - x^2}} + \arcsin x\).

Step 2: Apply the Chain Rule

The derivative of the second term, \(\sqrt{1 - x^2}\), can be found by applying the chain rule, which states that the derivative of a composed function is the derivative of the outer function times the derivative of the inner function evaluated at the original inner function. Here, \(\sqrt{u}\) is the outer function and \(1 - x^2\) is the inner function. The derivative of \(\sqrt{u}\) is \(\frac{1}{2\sqrt{u}}\) and the derivative of \(1 - x^2\) is \(-2x\). Thus, the derivative of the second term is: \(\frac{1}{2\sqrt{1 - x^2}}(-2x) = -\frac{x}{\sqrt{1 - x^2}}\).

Step 3: Combine the Results

Combining the derivatives found in the first and second steps gives the derivative of the entire function: \(\frac{x}{\sqrt{1 - x^2}} + \arcsin x - \frac{x}{\sqrt{1 - x^2}} = \arcsin x.\)