Calculating Partial Derivatives: A Step-by-Step Guide

Hey guys! Let's dive into the fascinating world of partial derivatives! This topic can seem a bit intimidating at first, but trust me, with a clear understanding and a few practice problems, you'll be acing these questions in no time. Today, we'll break down the question: Seja a função h(x, y, z) = 2x * sen(2y). Determine a soma de ∂h/∂x + ∂h/∂z no ponto (x, y, z) = (0, 0, 2). We'll find out the correct answer from the multiple-choice options, which are A -144, B 144, C -96, D 96, and E -48. Get ready to flex those math muscles!

Understanding Partial Derivatives: The Basics

Alright, before we jump into the problem, let's make sure we're all on the same page about what partial derivatives actually are. Imagine you've got a function with multiple variables – in our case, x, y, and z. A partial derivative is just the derivative of that function with respect to one of those variables, while treating all the other variables as constants. That's the key idea! Think of it like zooming in on how the function changes as you wiggle only one variable, holding all the others steady. For instance, when calculating the partial derivative of h with respect to x (written as ∂h/∂x), we treat y and z as constants. Similarly, when calculating ∂h/∂y, we treat x and z as constants, and so on. The process itself is pretty straightforward, and it boils down to applying the standard rules of differentiation that you already know. The most common of these include the power rule, the product rule, the chain rule, and the derivatives of trigonometric functions like sine and cosine. Make sure you're comfortable with these before you get started. Also, keep in mind that the point at which you need to evaluate the partial derivatives is always the last step.

So, what does it mean in practice? Let’s imagine a simple function like f(x, y) = x^2 + 3y. If we take the partial derivative of f with respect to x, we get ∂f/∂x = 2x. The 3y term vanishes because we treat y as a constant. If we take the partial derivative of f with respect to y, we get ∂f/∂y = 3, because the derivative of x^2 is zero. Remember that! It's super important to keep track of which variable you're differentiating with respect to and which ones you're treating as constants. This approach lets us analyze how a multivariable function changes in specific directions, which is super useful in all sorts of applications, from physics and engineering to economics and computer science. Think about it: If you want to know how a change in the price of one product affects the demand for another product, or how a change in temperature affects the rate of a chemical reaction, partial derivatives are your go-to tools.

Calculating ∂h/∂x

Alright, let’s get down to business and calculate the first part of our problem: ∂h/∂x. Remember, our function is h(x, y, z) = 2x * sen(2y). Now, we're taking the derivative with respect to x, which means we treat y and z as constants. The only term that involves x is 2x. The derivative of 2x with respect to x is simply 2. Because sen(2y) is considered a constant in this case, we just multiply it by the derivative of 2x, resulting in ∂h/∂x = 2 * sen(2y). That was simple, right?

Now, here comes the next step. To find the value of ∂h/∂x at the point (0, 0, 2), we need to plug in the values of x, y, and z. However, notice that x, y, and z don't directly appear in our expression for ∂h/∂x: it's only dependent on y. So, we substitute y = 0 into the equation ∂h/∂x = 2 * sen(2y), and we get ∂h/∂x = 2 * sen(2 * 0) = 2 * sen(0) = 2 * 0 = 0. Therefore, at the point (0, 0, 2), ∂h/∂x = 0. We've got the first part of our answer, guys!

Calculating ∂h/∂z

Now, let's figure out the second part of the question: calculating ∂h/∂z. Here's where things get super interesting. Remember, our function is still h(x, y, z) = 2x * sen(2y). But this time, we're taking the derivative with respect to z. Now, does the variable z appear anywhere in the function? Nope! Since h is not a function of z, it means that h is constant with respect to z. The derivative of a constant is always zero. Thus, ∂h/∂z = 0. No matter what values we substitute for x, y, and z, the partial derivative ∂h/∂z will always be zero.

So, at the point (0, 0, 2), ∂h/∂z = 0. That's a nice easy one! We are making great progress towards the final answer.

Finding the Sum ∂h/∂x + ∂h/∂z

Okay, guys, we're in the home stretch now. The question asks us to find the sum of ∂h/∂x + ∂h/∂z at the point (0, 0, 2). We've already calculated both partial derivatives at that point:

• ∂h/∂x = 0
• ∂h/∂z = 0

So, the sum is simply 0 + 0 = 0. Now, let’s go back to our multiple-choice options to find the correct answer! Looking at the options provided (A -144, B 144, C -96, D 96, and E -48), we see that none of them match our result, which is zero. However, we have made some calculation mistakes. Let's start over, with special emphasis on the evaluation point!

We have: h(x, y, z) = 2x * sen(2y).

• ∂h/∂x = 2 * sen(2y). At (0, 0, 2), ∂h/∂x = 2 * sen(2 * 0) = 2 * sen(0) = 2 * 0 = 0.

• ∂h/∂z = 0. At (0, 0, 2), ∂h/∂z = 0.

Then, the sum of ∂h/∂x + ∂h/∂z at (0, 0, 2) is 0 + 0 = 0. Since none of the answers match our result, let's analyze the problem again. I believe we have made an error in the statement of the problem and the answer options may be wrong.

Conclusion

So, there you have it! We've successfully navigated the world of partial derivatives and found the values for ∂h/∂x and ∂h/∂z, and determined their sum. Remember the key is to take your time, break the problem down into manageable steps, and always keep track of which variables are being held constant. Practice makes perfect, so keep working through problems, and you'll become a partial derivatives pro in no time! Keep in mind that we're dealing with the fun part of math here. Don't be afraid to experiment, explore, and most of all, have fun with it! If you enjoyed this explanation, and if you want to understand more about mathematical problems, let me know. I'm always happy to help.