15–48. Derivatives Find the derivative of the following functions.
y = 4^-x sin x

Verified step by step guidance

1. Step 1: Identify the function as a product of two functions, \(y = 4^{-x} \sin x\). This suggests using the product rule for differentiation.
2. Step 2: Recall the product rule for derivatives, which states that if \(y = u(x) \cdot v(x)\), then \(y' = u'(x) \cdot v(x) + u(x) \cdot v'(x)\). Here, let \(u(x) = 4^{-x}\) and \(v(x) = \sin x\).
3. Step 3: Differentiate \(u(x) = 4^{-x}\). Use the chain rule: \(u'(x) = \frac{d}{dx}(4^{-x}) = 4^{-x} \cdot \ln(4) \cdot (-1)\), which simplifies to \(-4^{-x} \ln(4)\).
4. Step 4: Differentiate \(v(x) = \sin x\). The derivative is straightforward: \(v'(x) = \cos x\).
5. Step 5: Apply the product rule: \(y' = (-4^{-x} \ln(4)) \cdot \sin x + 4^{-x} \cdot \cos x\). This expression represents the derivative of the given function.

Key Concepts

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that allows us to determine how a function behaves at any given point. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a specific point.

Product Rule

The Product Rule is a formula used to find the derivative of the product of two functions. It states that if you have two functions, u(x) and v(x), the derivative of their product is given by u'v + uv'. This rule is essential when differentiating functions that are multiplied together, such as in the given function y = 4^-x sin x.

Chain Rule

The Chain Rule is a method for differentiating composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x is the derivative of y with respect to u multiplied by the derivative of u with respect to x. This rule is particularly useful when dealing with functions that involve exponentials or trigonometric functions, as seen in the function y = 4^-x.