Finding The Derivative: A Calculus Problem

Hey math enthusiasts! Let's dive into a classic calculus problem. We're tasked with finding the value of the derivative, specifically f'(ln(-2)), given the function f(x) = 3x - 2e^(-x). This might seem a bit daunting at first, but trust me, we'll break it down step by step and make it super clear. The key here is understanding derivatives and how they work. We'll be using some basic rules of differentiation to tackle this problem, so let's get started!

Understanding the Problem: Derivatives and the Function

So, what exactly are we dealing with? We've got a function, f(x), and we want to find its derivative, denoted as f'(x). The derivative, in simple terms, tells us the rate of change of the function at any given point. Imagine a curve on a graph; the derivative at a specific point is essentially the slope of the tangent line at that point. In this case, our function f(x) = 3x - 2e^(-x), is a combination of a linear term (3x) and an exponential term (-2e^(-x)). Calculating f'(ln(-2)) means finding the slope of the tangent line to this function when x is equal to the natural logarithm of -2. But, wait a minute, the natural logarithm of a negative number? That's not defined in the real numbers, right? This is where things get interesting! We'll need to carefully consider this as we move forward with our calculations. Remember, the derivative of a sum or difference of terms is simply the sum or difference of the derivatives of those terms. The chain rule, a crucial concept in calculus, comes into play when differentiating composite functions like exponential functions with a power that's a function itself. Remember, in calculus, paying close attention to detail is crucial, especially when dealing with such nuances of domain and range. Getting the derivative is only half the battle, though. Then comes the evaluation: plugging in ln(-2) into the f'(x) we get. This requires us to interpret the result in the context of the domain of the original function. The domain will be essential since we are talking about ln(-2). Lastly, we will need to pick the best possible answer of A, B, C or D or maybe a totally new answer, and that is going to be the most satisfying part of our solution. So, let’s get right into the heart of the solution!

Step-by-Step Solution: Differentiating and Evaluating

Alright, let's roll up our sleeves and solve this. First, we need to find the derivative of f(x). We have f(x) = 3x - 2e^(-x). Let's find f'(x), term by term:

1. Derivative of 3x: The derivative of 3x with respect to x is simply 3. This is because the power rule states that the derivative of ax is a, where a is a constant.
2. Derivative of -2e^(-x): This part requires the chain rule. The derivative of e^u is e^u times the derivative of u with respect to x. Here, u = -x. So, the derivative of -x is -1. Thus, the derivative of -2e^(-x) is -2 * e^(-x) * (-1) = 2e^(-x).

So, putting it all together, we get f'(x) = 3 + 2e^(-x). Now, the tricky part. We need to evaluate f'(ln(-2)). As we mentioned earlier, the natural logarithm, ln(-2), is undefined in the real number system because the logarithm of a negative number is not a real number. This is where we need to think carefully about the implications. Our function itself is well-defined as far as derivatives are concerned. The derivative operation can still be performed using standard calculus rules. However, attempting to evaluate f'(x) at a value of x that is not in the domain of the original function's definition will not yield a real number. In fact, if we consider that the natural logarithm is only defined for positive numbers, then the entire question has a problem. Based on this, evaluating f'(ln(-2)) is not possible within the realm of real numbers. Therefore, the most accurate answer would be that the derivative at this point is undefined. The provided options are A) 1, B) 2, C) 5, D) ∅. Given the circumstances, the correct answer is D) ∅ because the natural logarithm of a negative number does not exist. The derivative exists, but its evaluation is not possible. To summarise, we first found the derivative of our function f(x), then we had to evaluate f'(ln(-2)), which is undefined. This brings us to the final conclusion that the answer is the empty set.

Conclusion: The Answer and Its Implications

So, after careful consideration, we've determined that f'(ln(-2)) is undefined in the real number system, making the correct answer D) ∅ (the empty set). This problem highlights the importance of understanding the domain of functions and the limitations of certain mathematical operations. Derivatives are powerful tools, but they must be applied with a thorough understanding of the underlying concepts.

This question can be a bit of a trick if you're not careful. The temptation is to go ahead and substitute ln(-2) into the derivative function without considering the fact that ln(-2) is not a real number. Always make sure to consider the domain and range of your functions. The answer D) ∅ emphasizes that you have to be careful when computing derivatives. Also, you must think about whether the computed derivative has a real-number value. If it doesn't, then you can't substitute it. This problem is a clever test of your understanding of derivatives and their limits. Always take a moment to understand each part of the problem. That way, you won't fall for the trick. Understanding the fundamentals of differentiation, including the chain rule, and the properties of logarithmic functions, is critical. With enough practice, these types of problems will become easier and even fun to solve! Good luck, and keep practicing!