Problem 13 - Calculus and its applications

Question

Differentiate. $$ g(x)=e^{2 x} $$

Step-by-Step Solution

Answer

The derivative of \( g(x) = e^{2x} \) is \( g'(x) = 2e^{2x} \).

Step 1: Identify the Function Type

The given function is an exponential function with base \( e \). Our task is to differentiate \( g(x) = e^{2x} \).

Step 2: Recall the Differentiation Rule

The derivative of \( e^{u} \), where \( u \) is a function of \( x \), is \( e^{u} \cdot \frac{du}{dx} \). Here, \( u = 2x \).

Step 3: Differentiate the Inner Function

Differentiate the exponent \( u = 2x \) with respect to \( x \). The derivative \( \frac{du}{dx} = 2 \).

Step 4: Apply the Chain Rule

Use the chain rule to differentiate \( g(x) = e^{2x} \) as follows: \( g'(x) = e^{2x} \cdot 2 \).

Step 5: Simplify the Expression

Simplify the expression to get the derivative: \( g'(x) = 2e^{2x} \).

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus used to find the derivative of complex functions by breaking them into simpler ones. It is particularly useful when dealing with compositions of functions. To put it simply, if you have two functions nested within each other, like \( f(g(x)) \), the chain rule tells us how to find the derivative.

• The general formula for the chain rule is \( (f(g(x)))' = f'(g(x)) \cdot g'(x) \), where \( f' \) and \( g' \) are the derivatives of \( f \) and \( g \) respectively.
• In our example, where \( g(x) = e^{2x} \), we have \( f(u) = e^u \) and \( u = 2x \). Applying the chain rule here requires us to differentiate \( e^u \) with respect to \( u \) and then multiply by the derivative of \( u = 2x \), which simplifies the whole differentiation process.

Understanding how to apply the chain rule correctly is crucial for success in calculus, as it is often the key step in finding derivatives of complex composite functions.

Exponential Function

Exponential functions are a class of mathematical functions that grow at an ever-increasing rate. These functions are expressed with the base of an exponential like \( e \) (approximately 2.71828), which is known as Euler's number, an essential constant in mathematics. An exponential function can be written in the form \( f(x) = a \cdot e^{bx} \), where:

• \( a \) is a constant that represents the initial amount or coefficient.
• \( e \) is the base of the natural logarithm.
• \( b \) is the growth factor in the exponent.

In our problem, \( g(x) = e^{2x} \), the function represents exponential growth where \( e \) is raised to a power determined by \( 2x \). This means every increase in \( x \) results in the function's value doubling at an exponential rate.

Derivative of Exponential Functions

Differentiating exponential functions can be straightforward if the right rules and techniques are understood. The nature of the exponential function, particularly when the base is \( e \), simplifies the differentiation process.

• The key differentiation rule one needs is that the derivative of \( e^u \), where \( u \) is any function of \( x \), is \( e^u \cdot \frac{du}{dx} \).
• This rule takes into account the application of the chain rule when the exponent is itself a function of \( x \).In our example, with \( u = 2x \), we first find \( \frac{du}{dx} = 2 \). Thus, the differentiation process becomes straightforward when you multiply \( e^{2x} \) by \( 2 \).

The result, \( g'(x) = 2e^{2x} \), is a clear demonstration of how powerful and efficient exponential differentiation can be once you understand the interplay between the chain rule and exponential function differentiation.