Problem 52 - Calculus of a Single Variable

Question

Find the derivative of the function. \(y=\frac{1}{2}\left[x \sqrt{4-x^{2}}+4 \arcsin \left(\frac{x}{2}\right)\right]\)

Step-by-Step Solution

Answer

The derivative of the original function \( y = \frac{1}{2}\left[x \sqrt{4-x^{2}}+4 \arcsin (\frac{x}{2})\right] \) is \( y' = \frac{2-x^{2}}{2\sqrt{4-x^{2}}} \)

Step 1: Identify Function Components

The function y can be broken down into two components: \( f(x) = \frac{1}{2}x \sqrt{4-x^{2}} \) and \( g(x) = \frac{1}{2} 4 \arcsin\left(\frac{x}{2}\right) \). We need to find their derivatives separately.

Step 2: Find Derivative of f(x)

The function \( f(x) \) requires the product and chain rule. First apply the product rule: \( (uv)' = u'v + uv' \). Here, \( u = x \) and \( v = \sqrt{4-x^{2}} \). So, \( u' = 1 \) and \( v' \) requires chain rule. For \( v' \), let \( h(x) = 4-x^{2} \), so \( v = \sqrt{h(x)} \). Applying chain rule for v: \( v' = \frac{1}{2\sqrt{h(x)}}* h'(x) = -\frac{x}{\sqrt{4-x^{2}}} \). Now apply the product rule to get \( f'(x) = \frac{1}{2} [\sqrt{4 - x^2} + x*(-\frac{x}{\sqrt{4-x^{2}}})] = \frac{1}{2}( \sqrt{4 - x^{2}} - x^{2}/\sqrt{4-x^{2}}) \)

Step 3: Find Derivative of g(x)

The function \( g(x) \) involves arcsin, so its derivative will be \( g'(x) = \frac{1}{2} * 4 * \frac{1}{\sqrt{1 - (x/2)^{2}}} * \frac{1}{2} = \frac{1}{\sqrt{4-x^{2}}} \)

Step 4: Combine Derivatives

The derivatives of \( f(x) \) and \( g(x) \) should be added to get the derivative of the entire function \( y = f(x) + g(x) \). So \( y' = f'(x) + g'(x) = \frac{2-x^{2}}{2\sqrt{4-x^{2}}} \)