To differentiate the expression \( 4^{-\frac{x}{2}} \cdot \sin(2x) \), we will use the product rule, chain rule, and logarithmic differentiation where necessary.

The product rule states that for two functions \( f(x) \) and \( g(x) \), the derivative is:

\[
\frac{d}{dx} \left[ f(x) \cdot g(x) \right] = f'(x) \cdot g(x) + f(x) \cdot g'(x)
\]

Here, let:
- \( f(x) = 4^{-\frac{x}{2}} \)
- \( g(x) = \sin(2x) \)

### Step 1: Differentiate \( f(x) = 4^{-\frac{x}{2}} \)
We rewrite the exponential part:
\[
4^{-\frac{x}{2}} = \left(2^2\right)^{-\frac{x}{2}} = 2^{-x}
\]
Now, differentiate \( 2^{-x} \) using the chain rule:
\[
\frac{d}{dx} 2^{-x} = 2^{-x} \ln(2) (-1) = -2^{-x} \ln(2)
\]
Thus:
\[
\frac{d}{dx} \left( 4^{-\frac{x}{2}} \right) = -2^{-x} \ln(2)
\]

### Step 2: Differentiate \( g(x) = \sin(2x) \)
By the chain rule:
\[
\frac{d}{dx} \sin(2x) = \cos(2x) \cdot \frac{d}{dx}(2x) = 2 \cos(2x)
\]

### Step 3: Apply the Product Rule
Using the product rule:
\[
\frac{d}{dx} \left( 4^{-\frac{x}{2}} \cdot \sin(2x) \right) = \left( -2^{-x} \ln(2) \right) \cdot \sin(2x) + 4^{-\frac{x}{2}} \cdot 2 \cos(2x)
\]

Simplifying:
\[
\frac{d}{dx} \left( 4^{-\frac{x}{2}} \cdot \sin(2x) \right) = -2^{-x} \ln(2) \sin(2x) + 2 \cdot 4^{-\frac{x}{2}} \cos(2x)
\]

This is the derivative of the expression.

Would you like a more detailed explanation of any step? Here are five questions to consider next:

1. How does the chain rule apply to exponential functions?
2. Why do we use logarithmic differentiation for powers of constants?
3. Can you generalize the product rule for more than two functions?
4. How would you apply this to other trigonometric products with exponentials?
5. What happens when \( x = 0 \) in this derivative?

**Tip:** Always simplify exponential expressions when differentiating, as it often leads to easier calculations!

Learn how to differentiate the expression 4^(-x/2) * sin(2x) using the product rule and chain rule. This solution includes step-by-step guidance, covering key concepts like exponential functions and trigonometric differentiation. Suitable for students studying undergraduate calculus.

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