Understanding the Derivative of e^(2x): A Step-by-Step Guide
The exponential function, particularly e^x (where 'e' is Euler's number, approximately 2.718), is a cornerstone of calculus and appears frequently in various scientific and engineering applications. Understanding its derivatives is crucial for solving problems related to growth, decay, and oscillations. This article focuses on finding the derivative of e^(2x), a slightly more complex variation, breaking down the process into manageable steps.
1. The Chain Rule: Your Essential Tool
Before tackling e^(2x), let's refresh our understanding of the chain rule. The chain rule is used to differentiate composite functions – functions within functions. It states:
If y = f(g(x)), then dy/dx = f'(g(x)) g'(x).
In simpler terms, we differentiate the "outer" function, leaving the "inner" function alone, then multiply by the derivative of the "inner" function.
2. Deconstructing e^(2x)
The function e^(2x) is a composite function. We can see this by identifying the inner and outer functions:
Our goal is to find the derivative of y = e^(2x) using the chain rule.
3. Applying the Chain Rule
Let's apply the chain rule step-by-step:
1. Differentiate the outer function: The derivative of e^u with respect to u is simply e^u. So, f'(u) = e^u.
2. Substitute the inner function: Replace 'u' with the inner function, 2x: f'(g(x)) = e^(2x).
3. Differentiate the inner function: The derivative of 2x with respect to x is 2. So, g'(x) = 2.
4. Multiply the results: According to the chain rule, we multiply the derivatives of the outer and inner functions: dy/dx = e^(2x) 2.
5. Final Result: Therefore, the derivative of e^(2x) is 2e^(2x).
4. Practical Examples
Let's illustrate this with some practical examples:
Example 1: Find the instantaneous rate of change of population growth modeled by P(t) = 1000e^(0.05t) at t = 10 years.
First, find the derivative: P'(t) = 1000 0.05 e^(0.05t) = 50e^(0.05t).
Then, substitute t = 10: P'(10) = 50e^(0.0510) = 50e^(0.5) ≈ 82.43.
This means the population is growing at a rate of approximately 82.43 individuals per year at t=10 years.
Example 2: Find the slope of the tangent line to the curve y = e^(2x) at x = 0.
The derivative gives us the slope of the tangent line at any point. We already know the derivative is 2e^(2x).
Substitute x = 0: Slope = 2e^(20) = 2e^0 = 2.
The slope of the tangent line at x = 0 is 2.
5. Key Takeaways
The derivative of e^(2x) is 2e^(2x). This is a direct application of the chain rule. Remember to identify the inner and outer functions correctly, differentiate them separately, and then multiply the results. Mastering the chain rule is crucial for differentiating a wide range of composite functions.
FAQs
1. What if the exponent was something other than 2x, like 3x or ax? The derivative would follow the same pattern. For e^(ax), the derivative would be ae^(ax).
2. Why is the derivative of e^x equal to e^x? This is a fundamental property of the exponential function e^x. It's the only function whose derivative is itself.
3. Can I use the product rule instead of the chain rule here? No. The product rule applies when you have a product of functions, not a composition of functions like e^(2x).
4. What are some real-world applications of this derivative? It's used extensively in modeling exponential growth and decay in fields like population dynamics, radioactive decay, and compound interest calculations.
5. How can I practice more on this topic? Practice differentiating various composite functions involving the exponential function. Work through problems with different exponents and try to apply the chain rule methodically. Consult textbooks, online resources, or your instructor for more practice problems.
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