Problem 33 - Calculus Early Transcendentals

Question

Derivatives 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 function is \(f'(x) = 8x \ln(2x)(2x)^{4x}\).

Step 1: Identify the outer and inner functions

First, let's identify the outer and inner functions: - Outer function: \(g(u) = u^{4x}\) - Inner function: \(h(x) = 2x\) Our goal is to find the derivative of \(f(x) = (2x)^{4x} = g(h(x))\). To do so, we'll apply the chain rule: \(f'(x) = g'(h(x)) \cdot h'(x)\).

Step 2: Find the derivatives of the outer and inner functions

Find the derivatives of the outer and inner functions: - The derivative of the inner function can be found using the power rule, \(h'(x) = 2\). - The derivative of the outer function is a bit more complicated. We need to differentiate \(g(u) = u^{4x}\) with respect to \(x\). To do so, let's rewrite the function using logarithmic differentiation, \(\ln(g(u)) = 4x \ln(u)\). Now differentiate both sides with respect to \(x\): \(\frac{g'(u)}{g(u)} = 4 \ln(u)\), thus \(g'(u) = 4u \ln(u)\cdot g(u)\).

Step 3: Apply the chain rule

Now that we have the derivatives of the outer and inner functions, let's apply the chain rule to find \(f'(x) = g'(h(x)) \cdot h'(x)\). So, \(f'(x) = \left[4(2x)\ln(2x)(2x)^{4x}\right] \cdot 2\) Simplify the expression, \(f'(x) = 8x \ln(2x)(2x)^{4x}\). Therefore, the derivative of the given function is \(f'(x) = 8x \ln(2x)(2x)^{4x}\).

Key Concepts

Chain Rule

The chain rule is an essential tool in calculus for finding the derivative of composite functions. It allows us to differentiate a function made up of two or more simpler functions. In the context of the exercise, we have a composite function where one function is plugged inside another function.

When we have a function like \(f(x) = (2x)^{4x}\), it is the combination of two functions: the outer function \(g(u) = u^{4x}\) and the inner function \(h(x) = 2x\). To find the derivative, using the chain rule is crucial. The chain rule formula is:

• \(f'(x) = g'(h(x)) \cdot h'(x)\)

This means we first differentiate the outer function \(g(u)\) with respect to \(u\), then differentiate the inner function \(h(x)\) with respect to \(x\), and finally, multiply these results together.

The power of the chain rule lies in its ability to break down complex problems into simpler parts, making calculus more manageable.

Logarithmic Differentiation

Logarithmic differentiation is a technique used when dealing with more complicated functions, particularly those involving exponents. In this exercise, it helps with differentiating the outer function \(g(u) = u^{4x}\).

To use logarithmic differentiation:

• Take the natural logarithm (\( \ln \)) of both sides of the equation \(g(u) = u^{4x}\), resulting in \(\ln(g(u)) = 4x \ln(u)\).
• Differentiate both sides with respect to \(x\). Remember, when you differentiate \( \ln(g(u)) \), you'll need to use implicit differentiation: \( \frac{g'(u)}{g(u)} = 4 \ln(u)\).
• Solve for \(g'(u)\) by rearranging the equation: \(g'(u) = 4u \ln(u) \cdot g(u)\).

Logarithmic differentiation is especially useful when functions are composed of multiple layers that involve power and exponential terms. It simplifies the differentiation process significantly.

Power Rule

The power rule is one of the simplest and most commonly used differentiation techniques. It is used when differentiating functions of the form \(x^n\), where \(n\) is a constant.

In this problem:

• The power rule helps us find the derivative of the inner function \(h(x) = 2x\), making \(h'(x) = 2\).

To use the power rule:

• If you have a function \(f(x) = x^n\), then the derivative is \(f'(x) = nx^{n-1}\).

It is important to note that the power rule applies when the base is a variable raised to a constant power. When differentiating more complex functions with variable exponent or base like those in this exercise, using additional rules like the chain rule or logarithmic differentiation is necessary. The power rule remains a foundational concept to build more advanced differentiation techniques.