Problem 69 - Essential Calculus Early Transcendentals

Question

If \(f(x)=e^{2 x},\) find a formula for \(f^{(n)}(x)\)

Step-by-Step Solution

Answer

The \(n^{th}\) derivative of \(f(x) = e^{2x}\) is \(f^{(n)}(x) = 2^n e^{2x}\).

Step 1: Understand the function

We are given the function \(f(x) = e^{2x}\). Our task is to find a general formula for its \(n^{th}\) derivative, denoted as \(f^{(n)}(x)\).

Step 2: Calculate the first derivative

To find the first derivative, apply the chain rule. The derivative of \(f(x) = e^{2x}\) is given by:\[f'(x) = 2e^{2x}.\]

Step 3: Calculate the second derivative

Use the result from Step 2 to find the second derivative. Again, apply the chain rule to \(f'(x) = 2e^{2x}\).\[f''(x) = 4e^{2x}.\]

Step 4: Calculate the third derivative

Continuing the pattern from the previous steps, the third derivative is:\[f'''(x) = 8e^{2x}.\]

Step 5: Observe the pattern

Notice a pattern in the coefficients of the derivatives:- The first derivative has the coefficient \(2\).- The second derivative has the coefficient \(4 = 2^2\).- The third derivative has the coefficient \(8 = 2^3\).The pattern suggests that the \(n^{th}\) derivative has the coefficient \(2^n\).

Step 6: Write the general formula

Based on the observed pattern, the formula for the \(n^{th}\) derivative is:\[f^{(n)}(x) = 2^n e^{2x}\].

Key Concepts

Chain Rule

The chain rule is a fundamental tool in calculus, used to differentiate functions that are compositions of other functions. In simpler terms, it's a method to find the derivative of a function based on its nested structures.
In this exercise, the function you want to differentiate is \(f(x) = e^{2x}\). Here, the inner function is \(2x\) and the outer function is the exponential \(e^{u}\).

• First, calculate the derivative of the outer function with respect to its inner function. For \(e^{u}\), this derivative remains \(e^{u}\).
• Next, find the derivative of the inner function \(2x\), which is simply \(2\).
• Finally, multiply these results. Thus, applying the chain rule to \(f(x)\) gives \(f'(x) = 2e^{2x}\).

With mastery over the chain rule, complex functions can be tackled easily by breaking them down into simpler parts.

Exponential Functions

Exponential functions are characterized by their constant rate of change and a base raised to an exponent. The function \(f(x) = e^{2x}\) is an exponential function where the base is \(e\), the natural number approximately equal to 2.718281.
These functions grow rapidly and are self-similar, meaning the shape of their graph is the same no matter the scale.

• The derivative of an exponential function \(b^x\), where \(b\) is a constant, has a unique property: the derivative is proportional to the function itself.
• In \(f(x) = e^{2x}\), the derivative \(f'(x) = 2e^{2x}\) demonstrates this property, as it still involves \(e^{2x}\).
• This self-replicating course is why the exponential function is used so often in modeling growth and decay in calculus.

Understanding these features of exponential functions helps in predicting real-world exponential growth patterns effectively.

Differentiation Techniques

Differentiation is the process of finding the rate at which a function is changing at any point. It's one of the cornerstones of calculus. Different techniques are employed depending on the form of the function.
For functions like \(f(x) = e^{2x}\), differentiation involves repeated application of basic rules.

• The primary technique used here is recognizing patterns. Calculating the first few derivatives \(f'(x) = 2e^{2x}\), \(f''(x) = 4e^{2x}\), and \(f'''(x) = 8e^{2x}\) reveals a clear pattern, allowing predictions about higher-orders.
• Such observation underlies the process of finding the nth derivative: recognizing how coefficients change as powers of 2.
• In calculus, learning and recognizing patterns in derivatives can simplify the procedure significantly, as demonstrated in this exercise, ultimately leading to the formula: \(f^{(n)}(x) = 2^n e^{2x}\).

Familiarity with differentiation techniques, such as pattern recognition and rules application, is essential for handling a variety of mathematical challenges effectively.