Problem 26 - Calculus Single Variable

Question

Calculate the derivative with respect to \(x\) of the given expression. \(\ln \left(1+3^{x}\right)\)

Step-by-Step Solution

Answer

The derivative of \( \ln(1 + 3^x) \) with respect to \( x \) is \( \frac{3^x \ln 3}{1 + 3^x} \).

Step 1: Identify the Function and its Inner Function

The given function is \( y = \ln(1 + 3^x) \). It is a composite function, where the outer function is the natural logarithm \( \ln(u) \) and the inner function is \( u = 1 + 3^x \).

Step 2: Differentiate the Outer Function

The derivative of \( \ln(u) \) with respect to \( u \) is \( \frac{1}{u} \). Keep the inner function \( u = 1 + 3^x \) in mind for later steps.

Step 3: Differentiate the Inner Function

The inner function is \( u = 1 + 3^x \). The derivative of \( 3^x \) with respect to \( x \) is found using the rule \( \frac{d}{dx}[a^x] = a^x \ln a \). Thus, \( \frac{d}{dx}[3^x] = 3^x \ln 3 \). Since the derivative of 1 is 0, we have \( \frac{d}{dx}[u] = 3^x \ln 3 \).

Step 4: Step 4: Apply the Chain Rule

According to the chain rule, the derivative of a composite function \( y = \ln(u) \) where \( u = 1+3^x \), is given by \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \). We have \( \frac{dy}{du} = \frac{1}{u} = \frac{1}{1 + 3^x} \) and \( \frac{du}{dx} = 3^x \ln 3 \).

Step 5: Multiply the Derivatives

Multiply the derivatives obtained in the previous steps: \( \frac{dy}{dx} = \frac{1}{1 + 3^x} \cdot 3^x \ln 3 \).

Step 6: Simplify the Expression

Simplify the expression from the previous step: \( \frac{dy}{dx} = \frac{3^x \ln 3}{1 + 3^x} \). This is the derivative of the original function with respect to \( x \).

Key Concepts

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. Composite functions are functions made by combining two or more functions. The essence of the chain rule is to systematically differentiate each function within the composite function.Imagine a composite function as a series of nested boxes. To differentiate the entire function, you need to first differentiate the outer box and then work your way into the inner boxes. The formula for the chain rule states that if you have a composite function \( y = f(g(x)) \), then the derivative \( \frac{dy}{dx} \) is given by:

• \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \)

In this context:

• \( \frac{dy}{du} \) is the derivative of the outer function with respect to the inner function.
• \( \frac{du}{dx} \) is the derivative of the inner function with respect to \( x \).

By applying the chain rule, you can tackle complex differentiation problems step-by-step, leading to the final derivative result.

Composite Functions

Composite functions are like layers of an onion, where you have functions inside other functions. In math terms, they are functions that can be written in the form \( f(g(x)) \), meaning one function \( f \) is applied to the output of another function \( g \). This structure is essential when dealing with problems in calculus such as differentiating a function like \( \ln(1 + 3^x) \).Understanding composite functions is key to applying the chain rule effectively. In the example \( y = \ln(1 + 3^x) \), we see:

• The outer function is \( \ln(u) \).
• The inner function is \( u = 1 + 3^x \).

This breakdown allows us to tackle the derivative by taking each part of the composite separately and applying differentiation rules along the way. Recognizing and working with these layers is crucial for solving problems involving composite functions.

Derivative of Exponential Functions

Exponential functions, such as \( a^x \) where \( a \) is a constant, have a unique rule for differentiation. The derivative of an exponential function \( a^x \) is given by:

• \( \frac{d}{dx}[a^x] = a^x \ln a \)

This rule is very helpful when differentiating expressions that contain exponential terms, like in the given problem of finding the derivative of \( 3^x \).In our exercise, to find the derivative of the inner function \( 1 + 3^x \), we only need to apply the rule to \( 3^x \) as the derivative of a constant (like 1) is zero. Thus, the derivative of \( 3^x \) is:

• \( \frac{d}{dx}[3^x] = 3^x \ln 3 \)

This understanding of how to differentiate exponential functions is a vital tool in calculus, particularly when they appear as part of composite functions like in our worked example.