Problem 33 - Calculus Early Transcendentals

Question

Evaluate the derivatives of the following functions. $$f(x)=(2 x)^{4 x}$$

Step-by-Step Solution

Answer

Question: Find the derivative of the function $$f(x) = (2x)^{4x}$$. Answer: The derivative of the given function is $$f'(x) = (2x)^{4x}\left(4\ln{(2x)} + 2\right)$$.

Step 1: Rewrite the function using natural logarithm properties

Let's take the natural logarithm of the function $$f(x)=(2x)^{4x}$$. Then, we can apply the power rule of logarithms to rewrite the function in terms of exponents: $$\ln{f(x)} = \ln{(2x)^{4x}} = 4x\ln{(2x)}$$

Step 2: Differentiate with respect to x using implicit differentiation

Now we can differentiate both sides of the equation with respect to x using implicit differentiation. Keep in mind that we're trying to find $$\frac{d}{dx} (f(x))$$, so we'll need to apply the chain rule while differentiating the right-hand side of the equation: $$\frac{f'(x)}{f(x)} = 4\ln{(2x)} + 4x\cdot\frac{1}{2x}$$

Step 3: Simplify the expression

Now, we will simplify the expression on the right-hand side: $$\frac{f'(x)}{f(x)} = 4\ln{(2x)} + 2$$

Step 4: Solve for f'(x)

Since we're interested in finding $$f'(x)$$, let's multiply both sides of the equation by $$f(x) = (2x)^{4x}$$: $$f'(x) = (2x)^{4x}\left(4\ln{(2x)} + 2\right)$$ So, the derivative of the given function is: $$f'(x) = (2x)^{4x}\left(4\ln{(2x)} + 2\right)$$

Key Concepts

Natural Logarithm

The natural logarithm, often denoted as \( \ln{x} \), is a fundamental mathematical function, particularly in calculus. It is the logarithm to the base \( e \), where \( e \approx 2.71828 \). It allows us to simplify complex expressions, especially those involving powers and exponents.

In this exercise, the natural logarithm helps transform the expression \( (2x)^{4x} \) into a more workable form for differentiation. By applying the logarithm rules, we can rewrite the initial function \( \ln{((2x)^{4x})} = 4x\ln{(2x)} \).

This transformation utilizes one of the logarithmic properties stating that \( \ln{(a^b)} = b\ln{a} \). Using logarithms in calculus helps manage the complexity of derivatives that involve exponential functions, simplifying the differentiation process.

Power Rule

The Power Rule is a basic principle in calculus used to find the derivative of a power of a variable. It states that if \( f(x) = x^n \), the derivative \( f'(x) \) is given by \( nx^{n-1} \). This rule simplifies the process of finding derivatives by providing a direct method of computing them for polynomial expressions.

In our specific example of differentiating \( (2x)^{4x} \), we indirectly use the Power Rule in conjunction with the properties of logarithms. By taking the natural logarithm, we manage the power term \( 4x \), allowing us to simplify and directly apply differentiation strategies. Although it might not seem evident at first, appreciating the Power Rule's mechanics is vital when preparing to handle complex derivatives involving polynomial, logarithmic, or exponential forms.

Implicit Differentiation

Implicit differentiation is a technique used when a function is not easily expressed as \( y = f(x) \) but is instead given implicitly. It enables us to find the derivative of such functions by differentiating both sides of an equation with respect to \( x \), and then solving for \( \frac{dy}{dx} \).

This exercise demonstrates implicit differentiation where we handle the equation \( \ln{f(x)} = 4x\ln{(2x)} \). We differentiate both sides to find \( \frac{f'(x)}{f(x)} = 4\ln{(2x)} + 4 \cdot \frac{x}{2x} \). By applying the chain rule effectively, we carefully obtain \( f'(x) \), ensuring each differentiation step respects implicit relationships within the function.

Implicit differentiation is indispensable for tackling equations where solving algebraically might not be feasible, streamlining the process of derivative calculation in a broad range of mathematical problems.