7–28. Derivatives Evaluate the following derivatives.


d/dx ((2x)⁴ˣ)

Verified step by step guidance

1. Step 1: Recognize that the given function is of the form f(x) = (2x)^(4x), which involves both a base and an exponent that are functions of x. This requires the use of logarithmic differentiation to simplify the derivative.
2. Step 2: Take the natural logarithm of both sides to simplify the expression. Let y = (2x)^(4x), then ln(y) = ln((2x)^(4x)). Using logarithmic properties, rewrite this as ln(y) = 4x * ln(2x).
3. Step 3: Differentiate both sides of the equation with respect to x. For the left-hand side, use implicit differentiation: d/dx[ln(y)] = (1/y) * dy/dx. For the right-hand side, apply the product rule to differentiate 4x * ln(2x).
4. Step 4: Apply the product rule to differentiate 4x * ln(2x). The derivative of 4x is 4, and the derivative of ln(2x) is (1/(2x)) * 2 (using the chain rule). Combine these results to find the derivative of the right-hand side.
5. Step 5: Solve for dy/dx by multiplying through by y (which is (2x)^(4x)) to isolate dy/dx. Substitute back the original expression for y to express the derivative in terms of x.

Key Concepts

Derivatives

A derivative represents the rate of change of a function with respect to a variable. It is a fundamental concept in calculus that allows us to understand how a function behaves as its input changes. The notation d/dx indicates differentiation with respect to the variable x, and the derivative can be interpreted as the slope of the tangent line to the function's graph at a given 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 d(uv)/dx = u'v + uv'. This rule is essential when differentiating expressions where two functions are multiplied together, as is the case in the given problem.

Exponential Functions

Exponential functions are functions of the form f(x) = a^x, where a is a constant and x is the variable. In the context of derivatives, the derivative of an exponential function can be expressed in terms of the original function, particularly when the exponent itself is a function of x. Understanding how to differentiate exponential functions is crucial for solving problems involving expressions like (2x)⁴ˣ.