Problem 58 - Calculus and its applications

Question

Use the Chain Rule, implicit differentiation, and other techniques to differentiate each function given. $$ f(x)=3^{\left(2^{x}\right)} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 3^{2^x} \ln(3) \cdot 2^x \ln(2) \).

Step 1: Identify the Function Form

The function is given as a combination of exponential functions: \( f(x) = 3^{2^x} \). This involves an outer function \( 3^u \) where \( u = 2^x \) and an inner function \( 2^x \).

Step 2: Apply the Chain Rule

To differentiate \( f(x) = 3^{2^x} \), we need to apply the chain rule. The chain rule states that \( \frac{d}{dx}[g(h(x))] = g'(h(x)) \cdot h'(x) \). Here, \( g(u) = 3^u \), and \( h(x) = 2^x \).

Step 3: Differentiate the Outer Function

Differentiate the outer function \( g(u) = 3^u \) with respect to \( u \). The derivative is \( g'(u) = 3^u \ln(3) \).

Step 4: Differentiate the Inner Function

Differentiate the inner function \( h(x) = 2^x \) with respect to \( x \). The derivative is \( h'(x) = 2^x \ln(2) \).

Step 5: Apply Chain Rule to Find the Derivative of f(x)

Combine the results from differentiating the outer and inner functions using the chain rule: \[ f'(x) = g'(h(x)) \cdot h'(x) = 3^{2^x} \ln(3) \cdot 2^x \ln(2). \]

Step 6: Simplify the Expression

Simplify the resulting expression for the derivative: \[ f'(x) = (3^{2^x})(\ln(3))(2^x)(\ln(2)). \] This is the derivative of the given function.

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used when dealing with equations that define one variable implicitly as a function of another variable. Unlike explicit functions, where one variable is directly expressed in terms of the other, implicit functions mix the variables together, making direct differentiation challenging. Instead of solving the equation explicitly, we differentiate both sides of the equation with respect to the independent variable and then solve for the derivative.

This method is particularly useful for functions that cannot be easily isolated or expressed in standard forms. For example, when dealing with equations like \( x^2 + y^2 = 1 \), which describes a circle, implicit differentiation allows us to find the derivative \( \frac{dy}{dx} \) without explicitly solving for \( y \).

To apply implicit differentiation, remember to:

• Differentiate both sides of the equation with respect to the independent variable.
• Apply the chain rule where necessary.
• Solve for the desired derivative, usually \( \frac{dy}{dx} \).

It's a powerful tool to have in your calculus toolkit, especially when functions are intertwined with each other.

Exponential Functions Differentiation

Differentiating exponential functions is straightforward but essential in calculus. An exponential function has the form \( a^x \), where the base \( a \) is a constant, and \( x \) is the variable. The key rule to remember when differentiating such functions is that the derivative of \( a^x \) with respect to \( x \) is \( a^x \ln(a) \). The natural logarithm \( \ln(a) \) is used because it relates the growth rate of the exponential function.

Consider the function \( g(x) = 3^{2^x} \), which is a composite function of an exponential function. To differentiate \( 3^{2^x} \), we use this rule combined with the chain rule. Here’s how it works step by step:

• First, identify the outer function, \( g(u) = 3^u \), with \( u = 2^x \).
• Differentiate the outer function with respect to \( u \) to get \( g'(u) = 3^u \ln(3) \).

The chain rule helps us tackle complex expressions, ensuring the derivative correctly accounts for each layer of the function.

Derivative of Composite Functions

When dealing with composite functions, such as \( f(x) = 3^{2^x} \), the chain rule is indispensable. Composite functions are formed when one function is applied to the result of another function. In this context, the expression can be simplified as \( g(h(x)) \), where \( g(u) = 3^u \) and \( h(x) = 2^x \).

The chain rule is a fundamental differentiation technique for finding the derivative of a composite function. It states that the derivative \( \frac{d}{dx}[g(h(x))] \) is the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function. Mathematically, it's expressed as:

• \( g'(h(x)) \cdot h'(x) \).

Using our function \( f(x) = 3^{2^x} \), the steps are:

• Differentiate the outer function \( g(u) = 3^u \) to get \( g'(u) = 3^u \ln(3) \).
• Differentiate the inner function \( h(x) = 2^x \) to obtain \( h'(x) = 2^x \ln(2) \).
• Apply the chain rule: \( f'(x) = 3^{2^x} \ln(3) \cdot 2^x \ln(2) \).

The chain rule elegantly combines different functions' derivatives, allowing us to solve problems that involve stacked or nested functions seamlessly.