Finding The Derivative: A Step-by-Step Guide

Hey everyone! Today, we're diving into the world of derivatives, specifically focusing on how to find the derivative of a function and evaluate it at a specific point. We'll be working through a classic calculus problem, breaking it down step by step to make sure everyone understands the process. So, let's get started!

Understanding the Problem: The Function and the Goal

Alright, guys, let's take a look at the function we're dealing with. We've got: $f(x) = \frac{5}{2x} - \frac{1}{3x^2}$. Our mission? To find $f'(1)$. In plain English, this means we need to find the derivative of the function $f(x)$ and then figure out what the value of that derivative is when x equals 1. Sounds simple enough, right? Well, it is! The key here is to remember the rules of differentiation and apply them systematically.

First, let's make sure we're all on the same page about what a derivative actually is. The derivative of a function represents the instantaneous rate of change of the function at a specific point. Think of it as the slope of the tangent line to the function's graph at that point. When we find $f'(1)$, we're finding the slope of the tangent line to the graph of $f(x)$ where $x = 1$. This concept is fundamental to calculus, providing insights into how a function is behaving at any given moment. This opens doors to a deeper understanding of rates of change, optimization problems, and more.

Now, let's break down the given function. We can rewrite it to make it easier to differentiate. Remember, it is useful to rewrite the original function. We'll rewrite the function using negative exponents to make differentiation simpler: $f(x) = \frac{5}{2}x^{-1} - \frac{1}{3}x^{-2}$. See how we've rewritten it? This form allows us to apply the power rule of differentiation directly. This is a very common trick, so keep it in mind. This transformation doesn't change the function's value; it just changes how it looks, which will help us when we take the derivative. Understanding these basic manipulations is key to success in calculus and a solid foundation for more complex problems.

Applying the Power Rule: Differentiating the Function

Okay, time for the fun part: finding the derivative! We're going to use the power rule, which is one of the most fundamental rules in calculus. The power rule states that if you have a function of the form $x^n$, its derivative is $nx^{n-1}$. Let's apply this to our function, term by term. We have two terms in our function, so we'll differentiate each separately.

For the first term, $(\frac{5}{2}x^{-1})$, we apply the power rule. The derivative is $\frac{5}{2} * (-1) * x^{-1-1}$, which simplifies to $-\frac{5}{2}x^{-2}$. Remember that the constant factor $\frac{5}{2}$ simply stays there during the differentiation, so we just multiply it by the derivative of $x^{-1}$.

Now, let's look at the second term, $\frac1}{3}x^{-2}$. Applying the power rule, we get $-\frac{1}{3} * (-2) * x^{-2-1}$, which simplifies to $\frac{2}{3}x^{-3}$. Again, the constant factor $\frac{1}{3}$ is simply multiplied by the result of the differentiation. Therefore, the derivative of our function, $f'(x)$, is the sum of the derivatives of each term $f'(x) = -\frac{5{2}x^{-2} + \frac{2}{3}x^{-3}$. Notice the sign change when differentiating the second term because we multiplied by -2. Also, remember to handle constants. The power rule is a lifesaver, so make sure you understand it!

This is a critical step. If you make a mistake here, you'll be in trouble later on. So double-check your work!

Evaluating the Derivative: Finding f'(1)

Alright, we've found the derivative, $f'(x)$. Now we just need to evaluate it at $x = 1$. This means we're going to substitute 1 for every $x$ in our derivative. It's like plugging in a value into a formula.

So, let's plug it in! We have $f'(1) = -\frac{5}{2}(1)^{-2} + \frac{2}{3}(1)^{-3}$. Now, remember that any number raised to the power of -2 or -3 is still 1. So, this simplifies to $f'(1) = -\frac{5}{2} * 1 + \frac{2}{3} * 1$, which is $f'(1) = -\frac{5}{2} + \frac{2}{3}$. We're almost there!

To finish, we need to combine these two fractions. To do that, we need a common denominator. The least common denominator for 2 and 3 is 6. So, we'll rewrite each fraction with a denominator of 6. We get $\frac{-5}{2} = \frac{-15}{6}$ and $\frac{2}{3} = \frac{4}{6}$. Now, we can add them! $\frac{-15}{6} + \frac{4}{6} = \frac{-11}{6}$. Therefore, $f'(1) = -\frac{11}{6}$. And there we have it! We found the derivative of the function, and then we evaluated it at x = 1. Congratulations, guys!

Final Answer: The Simplified Fraction

So, the answer is: $f'(1) = -\frac{11}{6}$. This is already in the simplest form, as the numerator and denominator have no common factors other than 1. And that's all there is to it! We've found the derivative and evaluated it at the specified point. This process is applicable to a wide range of functions, so keep practicing!

This is the final answer, and it represents the slope of the tangent line to the original function at the point where x = 1. This value tells us how the function is changing at that specific point. The negative sign indicates that the function is decreasing at x = 1. Understanding the meaning of the derivative is as crucial as the calculation itself. The ability to interpret the result adds a layer of depth to your mathematical understanding.

Key Takeaways and Tips for Success

• Rewrite the function: Sometimes, rewriting the function using exponents can make differentiation much easier. For example, rewriting fractions as negative exponents. This will save you a ton of time and effort! Make sure you're comfortable with this step.
• Power Rule Mastery: The power rule is your best friend. Practice it until you can apply it in your sleep. It's the bread and butter of differentiation.
• Be Careful with Signs and Constants: Keep a close eye on your signs and constants. A small mistake can lead to the wrong answer. Take your time, and double-check your work. This is where most errors happen.
• Practice, Practice, Practice: The more problems you work through, the better you'll get. Try different functions and different points. Repetition is key to mastering this concept. Find as many problems as possible and repeat the same steps. This will make you an expert in no time!
• Understand What It Means: Don't just focus on the calculations. Try to understand what the derivative represents graphically. This will deepen your understanding and make the process more intuitive.

By following these steps and tips, you'll be well on your way to mastering derivatives! Calculus might seem intimidating at first, but with practice and a good understanding of the basics, you'll be tackling these problems with confidence. Keep up the great work, and remember to always ask questions if you get stuck. Happy differentiating, everyone!