Finding The Derivative Of Y = 2 - Cos 2x A Step-by-Step Guide

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Hey guys! Today, we're diving into a super common problem in calculus: finding the derivative of a function. Specifically, we're going to tackle the function y = 2 - cos 2x. It might seem a bit intimidating at first, but trust me, we'll break it down step by step so it's easy to understand. This is crucial stuff if you're studying calculus, as derivatives pop up everywhere, from physics to economics. So, let’s get started and figure out how to find the derivative of this function!

Understanding Derivatives

First off, let's quickly recap what a derivative actually is. In simple terms, the derivative of a function tells you its rate of change at any given point. Think of it like the slope of a curve at a specific spot. If the derivative is positive, the function is increasing; if it's negative, it's decreasing; and if it's zero, you're at a peak or a trough. Knowing how to find derivatives is super important because it allows us to analyze how functions behave and make predictions based on that behavior. It's like having a superpower in the world of math and science!

The Basic Rules of Differentiation

Before we jump into our specific problem, we need to have a few basic rules of differentiation under our belts. These are the building blocks that will allow us to tackle more complex derivatives. Here are a couple of key rules:

  1. The Constant Rule: The derivative of a constant is always zero. So, if you have a function like y = 5, its derivative is 0. Simple, right?
  2. The Power Rule: If you have a function in the form y = x^n, its derivative is y' = n * x^(n-1). This is one of the most frequently used rules, so make sure you've got it down. For example, if y = x^3, then y' = 3x^2.
  3. The Constant Multiple Rule: If you have a constant multiplied by a function (like y = k * f(x)), the derivative is y' = k * f'(x). Basically, you just bring the constant along for the ride.
  4. The Sum/Difference Rule: If you're differentiating a sum or difference of functions (like y = f(x) + g(x) or y = f(x) - g(x)), you can differentiate each term separately. So, y' = f'(x) + g'(x) or y' = f'(x) - g'(x).
  5. The Chain Rule: This is where things get a little trickier, but it's essential for functions like the one we're dealing with today. If you have a composite function (like y = f(g(x))), the derivative is y' = f'(g(x)) * g'(x). In other words, you differentiate the outer function, keeping the inner function the same, and then multiply by the derivative of the inner function. This is crucial for handling the cos 2x part of our problem. Mastering the chain rule is vital for success in calculus!

Differentiation of Trigonometric Functions

We also need to know the derivatives of trigonometric functions, specifically cosine. The derivative of cos(x) is -sin(x). Remember this! It's a fundamental rule that we'll use directly in our problem. But what about cos(2x)? That’s where the chain rule comes in handy.

Now that we've refreshed our memory on these rules, we're ready to tackle our main problem: finding the derivative of y = 2 - cos 2x. Let's do this! Understanding these rules is half the battle, so if you're feeling shaky on any of them, take a moment to review before moving on.

Step-by-Step Solution for y = 2 - cos 2x

Okay, let's dive into finding the derivative of y = 2 - cos 2x. We'll take it one step at a time to make sure we don't miss anything. It’s like following a recipe – if you follow each step carefully, you'll get the right result. So, let’s break it down!

Step 1: Break Down the Function

First, we need to see the function as a sum or difference of simpler parts. In our case, y = 2 - cos 2x is the difference between a constant (2) and a trigonometric function (cos 2x). This makes it easier to apply our differentiation rules. Recognizing the structure of the function is the first key step. It's like looking at a complex puzzle and figuring out the individual pieces before trying to put them together. This initial breakdown helps us apply the right rules.

Step 2: Apply the Sum/Difference Rule

As we discussed earlier, the sum/difference rule allows us to differentiate each term separately. So, we'll differentiate 2 and -cos 2x individually. This simplifies the problem into smaller, more manageable parts. It’s like dividing a big task into smaller sub-tasks – each sub-task is easier to handle. Applying the sum/difference rule is a fundamental technique in differentiation.

Step 3: Differentiate the Constant Term

The derivative of the constant term, 2, is simply 0. Remember, the derivative of any constant is always zero because constants don't change. They're, well, constant! This is a straightforward application of the constant rule. Easy peasy! Constant differentiation is a basic rule, but it’s essential to remember.

Step 4: Differentiate -cos 2x

This is where the chain rule comes into play. We have a composite function here: the outer function is -cos(u) and the inner function is u = 2x. Remember the chain rule: if y = f(g(x)), then y' = f'(g(x)) * g'(x). So, we need to differentiate the outer function while keeping the inner function intact, and then multiply by the derivative of the inner function. The chain rule is the heart of this problem, so let’s go through it carefully.

  • Derivative of the Outer Function: The derivative of -cos(u) with respect to u is sin(u). Remember, the derivative of cos(x) is -sin(x), so the derivative of -cos(x) is -(-sin(x)) = sin(x). Got it? Understanding trigonometric derivatives is crucial here.
  • Derivative of the Inner Function: The inner function is u = 2x. The derivative of 2x with respect to x is 2. This is a simple application of the power rule (x^1) and the constant multiple rule. Inner function differentiation is a key part of the chain rule.
  • Apply the Chain Rule: Now, we multiply the derivative of the outer function (sin(u)) by the derivative of the inner function (2). This gives us 2sin(u). Substituting u = 2x back in, we get 2sin(2x). We’re getting closer! Chain rule application involves multiple steps, so take your time to ensure you understand each one.

Step 5: Combine the Results

Now, we combine the results from our individual differentiations. We found that the derivative of 2 is 0 and the derivative of -cos 2x is 2sin 2x. Adding these together, we get:

y' = 0 + 2sin 2x = 2sin 2x

And that’s it! We've found the derivative of y = 2 - cos 2x. The final combination of the results gives us our answer. This step-by-step approach makes the problem much more manageable.

The Answer and Why It Matters

So, the derivative of y = 2 - cos 2x is 2 sin 2x. If you were presented with multiple-choice options like A) 2 sin 2x, B) sin 2x, C) 4 cos 2x, D) -sin 2x, the correct answer is A) 2 sin 2x. Woohoo! We got there!

Why Does This Matter?

You might be thinking,