In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
3. y = (1/4)xe^(4x) - (1/16)e^(4x)

Verified step by step guidance

1. Identify the function to differentiate: \(y = \frac{1}{4} x e^{4x} - \frac{1}{16} e^{4x}\).
2. Recognize that the derivative of \(y\) with respect to \(x\) requires using the product rule for the first term \(\frac{1}{4} x e^{4x}\) and the chain rule for the exponential terms.
3. Apply the product rule to the first term: if \(u = \frac{1}{4} x\) and \(v = e^{4x}\), then \(\frac{d}{dx}(uv) = u'v + uv'\). Compute \(u' = \frac{1}{4}\) and \(v' = 4 e^{4x}\) using the chain rule.
4. Differentiate the second term \(- \frac{1}{16} e^{4x}\) using the chain rule: \(\frac{d}{dx} e^{4x} = 4 e^{4x}\), so multiply by the constant \(-\frac{1}{16}\).
5. Combine the derivatives from both terms to write the full expression for \(\frac{dy}{dx}\) before simplifying.

Key Concepts

Derivative of Exponential Functions

The derivative of an exponential function with base e, such as e^(kx), is found by applying the chain rule. Specifically, d/dx[e^(kx)] = k * e^(kx), where k is a constant. This rule is essential for differentiating terms like e^(4x).

Product Rule

The product rule is used to differentiate functions that are products of two differentiable functions. If y = u(x)v(x), then y' = u'v + uv'. This rule applies to terms like (1/4)x * e^(4x), where both factors depend on x.

Constant Multiple Rule

The constant multiple rule states that the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. For example, d/dx[c * f(x)] = c * f'(x). This simplifies differentiation of terms like (1/4)xe^(4x) and (1/16)e^(4x).