Unraveling the Derivative of e<sup>ln x</sup>: A Comprehensive Guide
This article aims to provide a thorough understanding of the derivative of the function e<sup>ln x</sup>. While seemingly complex, this seemingly complex function simplifies significantly using fundamental rules of calculus and the properties of logarithms and exponential functions. We will explore the process step-by-step, clarifying the underlying principles and providing illustrative examples to solidify your comprehension.
1. Understanding the Components: e<sup>x</sup> and ln x
Before diving into the derivative, let's refresh our understanding of the core functions involved: e<sup>x</sup> and ln x.
e<sup>x</sup> (the exponential function): This is the exponential function with base e, where e is Euler's number (approximately 2.71828). Its derivative is remarkably simple: d(e<sup>x</sup>)/dx = e<sup>x</sup>. This means the derivative of e<sup>x</sup> is itself.
ln x (the natural logarithm): This is the logarithm to the base e. It's the inverse function of e<sup>x</sup>. This means that if e<sup>a</sup> = b, then ln b = a. The derivative of ln x is 1/x.
2. Applying the Chain Rule
The function e<sup>ln x</sup> is a composite function, meaning it's a function within a function. To find its derivative, we need to employ the chain rule. The chain rule states that the derivative of a composite function, f(g(x)), is f'(g(x)) g'(x).
The expression e<sup>ln x</sup> simplifies significantly due to the inverse relationship between e<sup>x</sup> and ln x. Remember that e<sup>ln x</sup> = x. Therefore, our derivative becomes:
d(e<sup>ln x</sup>)/dx = x (1/x) = 1
This surprisingly simple result demonstrates the power of understanding the interplay between exponential and logarithmic functions. The derivative of e<sup>ln x</sup> is simply 1.
4. Practical Example
Let's consider a practical application. Suppose we have a function representing the growth of a population: P(t) = e<sup>ln(100 + 2t)</sup>, where t represents time in years. To find the rate of population growth at t=5 years, we need to find the derivative of P(t) with respect to t.
Using the chain rule and our knowledge that the derivative of e<sup>ln x</sup> is 1, we can simplify:
Therefore, the population growth rate at t=5 years is 2 units per year.
5. Conclusion
The seemingly complicated function e<sup>ln x</sup> simplifies dramatically when we apply the chain rule and utilize the inverse relationship between the exponential and natural logarithm functions. The derivative of e<sup>ln x</sup> is consistently 1, highlighting the elegant interplay of these fundamental mathematical concepts. Understanding this relationship is crucial for tackling more complex problems in calculus and related fields.
Frequently Asked Questions (FAQs):
1. Is the derivative always 1, regardless of the expression inside the ln? No, the derivative is only 1 if the expression inside the ln is simply x. If it's a more complex function, you must apply the chain rule properly, as shown in the example above.
2. What if the base of the exponential function is not e? If the base is different from e, you cannot directly apply the simplification e<sup>ln x</sup> = x. You would need to use logarithmic properties and the chain rule.
3. Can we use this concept in other areas of mathematics? Yes, this understanding is vital in various fields, including differential equations, physics (modeling exponential growth/decay), and economics (compound interest calculations).
4. What happens if x is negative or zero? The natural logarithm (ln x) is only defined for positive values of x. Therefore, the function e<sup>ln x</sup> and its derivative are only defined for x > 0.
5. Are there any limitations to this simplification? The simplification e<sup>ln x</sup> = x holds true only when x > 0. For x ≤ 0, the natural logarithm is undefined. Therefore, the derivative, too, is undefined in these domains.
Note: Conversion is based on the latest values and formulas.
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