Problem 41 - Calculus of a Single Variable

Question

Find the derivative of the function. \(g(x)=3 \arccos \frac{x}{2}\)

Step-by-Step Solution

Answer

The derivative of the function \( g(x) = 3 \arccos \frac{x}{2} \) is \( g'(x) = -\frac{3}{2 \sqrt{1-\frac{x^2}{4}}} \).

Step 1: Recognize the function within the function

Here, \( \frac{x}{2} \) is a function within the arccosine function. This indicates that we need to use the chain rule.

Step 2: Recall the derivative of the arccosine function

The derivative of arc cosine of \( u \) is \( - \frac{1}{\sqrt{1 - u^2}} \). We have to apply this to our function.

Step 3: Use the chain rule

To differentiate \( g(x) = 3 \arccos \frac{x}{2} \), we leave the constant \(3\) alone and differentiate \( \arccos \frac{x}{2} \) using the chain rule. This gives us: \( g'(x) = -3 \cdot \frac{1}{2} \cdot \frac{1}{\sqrt{1- (\frac{x}{2})^2}} \). This simplifies to \( g'(x) = - \frac{3}{2 \sqrt{1- \frac{x^2}{4}}} \).