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Magazine Chapter 3: Problem 69
If \(f(x)=e^{2 x},\) find a formula for \(f^{(n)}(x)\)
Short Answer
Expert verified
The \(n^{th}\) derivative of \(f(x) = e^{2x}\) is \(f^{(n)}(x) = 2^n e^{2x}\).
Step by step solution
01
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 \(f^{(n)}(x)\).
02
Calculate the first derivative
To find the first derivative, apply the chain rule. The derivative of \(f(x) = e^{2x}\) is given by:\[f'(x) = 2e^{2x}.\]
03
Calculate the second derivative
Use the result from Step 2 to find the second derivative. Again, apply the chain rule to \(f'(x) = 2e^{2x}\).\[f''(x) = 4e^{2x}.\]
04
Calculate the third derivative
Continuing the pattern from the previous steps, the third derivative is:\[f'''(x) = 8e^{2x}.\]
05
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 \(2^n\).
06
Write the general formula
Based on the observed pattern, the formula for the \(n^{th}\) derivative is:\[f^{(n)}(x) = 2^n e^{2x}\].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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 \(e^{u}\).
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 \(f'(x) = 2e^{2x}\).
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.
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.
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.
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: \(f^{(n)}(x) = 2^n e^{2x}\).
