Problem 36 - College Algebra and Calculus: An Applied Approach

Question

Use the General Power Rule to find the derivative of the function. $$ y=2 \sqrt{4-x^{2}} $$

Step-by-Step Solution

Answer

The derivative of the function \(y=2 \sqrt{4-x^{2}}\) is \(\frac{dy}{dx} = - \frac{x}{\sqrt{4-x^2}} \).

Step 1: Rewrite the function

The function can be rewritten in the power form: \(y=2(4-x^{2})^{0.5}\).

Step 2: Apply the General Power Rule

We need to find \(\frac{dy}{dx}\). The derivative of \(y\) is found using the General Power Rule, which states that the derivative of \(y = u^n\) with respect to \(x\) is \( \frac{dy}{dx} = n * u^{n-1} * \frac{du}{dx}\). For our function where \(u = 4 - x^2 \) and \(n = 0.5\), \(\frac{dy}{dx} = 0.5*2*(4-x^2)^{-0.5}*(-2x)\).

Step 3: Simplify the Solution

After simplification, \(\frac{dy}{dx} = - \frac{x}{\sqrt{4-x^2}} \).