Problem 56 - Calculus of a Single Variable

Question

Find the derivative of the function. \(y=25 \arcsin \frac{x}{5}-x \sqrt{25-x^{2}}\)

Step-by-Step Solution

Answer

The derivative of the function \(y=25 \arcsin \frac{x}{5}-x \sqrt{25-x^{2}}\) is \(y'(x) = 5\frac{1}{\sqrt{25 - x^2}} - \sqrt{25 - x^2} + \frac{x^2}{\sqrt{25 - x^2}}\).

Step 1: Identify the parts of the function

Split the function into two parts as follows: \(f(x) = 25 \arcsin(\frac{x}{5})\) and \(g(x) = -x\sqrt{25 - x^2}\). These parts will be differentiated separately.

Step 2: Find the derivative of the first part - \(f(x)\)

The derivative of \(f(x) = 25 \arcsin(\frac{x}{5})\), using the chain rule, is \(f'(x) = 25\frac{1}{\sqrt{1 - \left(\frac{x}{5}\right)^2}} \cdot \frac{1}{5} = 5\frac{1}{\sqrt{25 - x^2}}\).

Step 3: Find the derivative of the second part - \(g(x)\)

The derivative of \(g(x)=-x \sqrt{25-x^2}\), using the product rule and chain rule, is \(g'(x) = -\sqrt{25 - x^2} - x\left(\frac{1}{2\sqrt{25 - x^2}} \cdot -2x\right) = -\sqrt{25 - x^2} + \frac{x^2}{\sqrt{25 - x^2}}\).

Step 4: Combine the derivatives

The derivative of the total function \(y(x) = f(x) + g(x)\) is given by the sum of the derivatives of \(f(x)\) and \(g(x)\), i.e., \(y'(x) = f'(x) + g'(x) = 5\frac{1}{\sqrt{25 - x^2}} - \sqrt{25 - x^2} + \frac{x^2}{\sqrt{25 - x^2}}\) .