Derivative Of Y = 2^((1-cos(x))/(1+cos(x))): Solved!

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Hey guys! Today, we're diving into a fun calculus problem: finding the derivative of the function y = 2^((1-cos(x))/(1+cos(x))). It looks a little intimidating at first, but don't worry, we'll break it down step by step. We'll be using a combination of the chain rule, quotient rule, and some trigonometric identities to get to the final answer. So, grab your pencils, and let's get started!

Understanding the Problem

Before we jump into the solution, let's clearly understand what we need to do. We have a function y, which is 2 raised to the power of a fraction involving cosine. Our goal is to find dy/dx, which represents the instantaneous rate of change of y with respect to x. Essentially, we want to know how y changes as x changes. This is a classic calculus problem, and mastering these techniques is super important for many fields in science and engineering.

Keywords: derivative, chain rule, quotient rule, trigonometric identities, calculus, dy/dx, rate of change, exponential function, cosine function

Breaking Down the Function

To tackle this problem effectively, we need to break down the function into smaller, more manageable parts. Notice that we have an exponential function with a rather complex exponent. The exponent itself is a fraction involving the cosine function. This means we'll likely need to use the chain rule multiple times. The chain rule, in essence, helps us differentiate composite functions, that is, functions within functions. It tells us that if we have y = f(g(x)), then dy/dx = f'(g(x)) * g'(x). This means we differentiate the outer function while keeping the inner function as is, and then we multiply by the derivative of the inner function. In our case, the outer function is the exponential function 2^u, and the inner function is u = (1-cos(x))/(1+cos(x)). Let's keep this in mind as we move forward. Recognizing these components is the first step in solving complex problems. Understanding the structure allows us to plan our approach and apply the relevant differentiation rules.

Step-by-Step Solution

Alright, let's get our hands dirty and find the derivative! We'll go through this step-by-step so you can follow along easily.

Step 1: Applying the Chain Rule

As we discussed, we'll start with the chain rule. Let's rewrite our function as:

y = 2^u, where u = (1-cos(x))/(1+cos(x))

Now, we need to find dy/du and du/dx. Let's start with dy/du. We know that the derivative of a^u (where a is a constant) is a^u * ln(a). So,

dy/du = 2^u * ln(2)

Now we need to find du/dx, which is the derivative of (1-cos(x))/(1+cos(x)). This looks like a job for the quotient rule!

Keywords: chain rule, exponential function, derivative, dy/du, du/dx, quotient rule, ln(2)

Step 2: Using the Quotient Rule

The quotient rule states that if we have a function u(x) = f(x)/g(x), then the derivative u'(x) is [f'(x)g(x) - f(x)g'(x)] / [g(x)]^2. In our case, f(x) = 1-cos(x) and g(x) = 1+cos(x). Let's find their derivatives:

f'(x) = sin(x) g'(x) = -sin(x)

Now, we can apply the quotient rule to find du/dx:

du/dx = [sin(x)(1+cos(x)) - (1-cos(x))(-sin(x))] / (1+cos(x))^2

Let's simplify this a bit. Distribute the sin(x) terms:

du/dx = [sin(x) + sin(x)cos(x) + sin(x) - sin(x)cos(x)] / (1+cos(x))^2

Notice that the sin(x)cos(x) terms cancel out, leaving us with:

du/dx = 2sin(x) / (1+cos(x))^2

Awesome! We've found du/dx. This was a crucial step, and using the quotient rule correctly is key to getting the right answer. Make sure you're comfortable with the quotient rule; it's a fundamental tool in calculus. Now we have both dy/du and du/dx, so we can move on to the next step.

Keywords: quotient rule, f(x), g(x), sin(x), cos(x), derivative, du/dx, simplification, (1+cos(x))^2

Step 3: Combining the Results

Now that we have dy/du and du/dx, we can use the chain rule to find dy/dx:

dy/dx = (dy/du) * (du/dx)

Substitute the expressions we found earlier:

dy/dx = [2^u * ln(2)] * [2sin(x) / (1+cos(x))^2]

Remember that u = (1-cos(x))/(1+cos(x)). Let's substitute that back in:

dy/dx = [2^((1-cos(x))/(1+cos(x))) * ln(2)] * [2sin(x) / (1+cos(x))^2]

This is the derivative, but we can simplify it further using some trigonometric identities.

Keywords: chain rule, dy/dx, substitution, u = (1-cos(x))/(1+cos(x)), 2^u, ln(2), 2sin(x) / (1+cos(x))^2

Step 4: Simplifying with Trigonometric Identities

To simplify the expression, we can use the identity:

1 + cos(x) = 2cos^2(x/2)

And also:

sin(x) = 2sin(x/2)cos(x/2)

Let's substitute these identities into our expression for dy/dx:

dy/dx = [2^((1-cos(x))/(1+cos(x))) * ln(2)] * [2 * 2sin(x/2)cos(x/2) / (2cos2(x/2))2]

dy/dx = [2^((1-cos(x))/(1+cos(x))) * ln(2)] * [4sin(x/2)cos(x/2) / 4cos^4(x/2)]

We can cancel out the 4s and one cos(x/2) term:

dy/dx = [2^((1-cos(x))/(1+cos(x))) * ln(2)] * [sin(x/2) / cos^3(x/2)]

Now, let's use another trigonometric identity:

tan(x/2) = sin(x/2) / cos(x/2)

And:

sec(x/2) = 1 / cos(x/2)

So we can rewrite our expression as:

dy/dx = 2^((1-cos(x))/(1+cos(x))) * ln(2) * tan(x/2) * sec^2(x/2)

Now, let's simplify the exponent (1-cos(x))/(1+cos(x)) using the half-angle identities:

(1 - cos(x)) = 2sin^2(x/2) (1 + cos(x)) = 2cos^2(x/2)

So, (1-cos(x))/(1+cos(x)) = 2sin^2(x/2) / 2cos^2(x/2) = tan^2(x/2)

Substitute this back into the expression:

dy/dx = 2(tan2(x/2)) * ln(2) * tan(x/2) * sec^2(x/2)

This is a much more simplified form of the derivative. Isn't that satisfying?

Keywords: trigonometric identities, 1 + cos(x) = 2cos^2(x/2), sin(x) = 2sin(x/2)cos(x/2), tan(x/2), sec(x/2), half-angle identities, simplification, 2(tan2(x/2)), ln(2), tan(x/2) * sec^2(x/2)

Final Answer

So, after all that work, the derivative of y = 2^((1-cos(x))/(1+cos(x))) is:

dy/dx = 2(tan2(x/2)) * ln(2) * tan(x/2) * sec^2(x/2)

Woohoo! We did it! This problem was a great exercise in using the chain rule, quotient rule, and trigonometric identities. It might have seemed tough at the beginning, but breaking it down into smaller steps made it much more manageable. Remember, practice makes perfect, so keep tackling those calculus problems!

Keywords: final answer, derivative, 2(tan2(x/2)) * ln(2) * tan(x/2) * sec^2(x/2), calculus, chain rule, quotient rule, trigonometric identities, practice

Key Takeaways

Let's recap some of the key concepts we used in solving this problem:

  1. Chain Rule: Remember to use the chain rule when differentiating composite functions (functions within functions). It's crucial for handling problems like this where we have an exponential function raised to a complex power.
  2. Quotient Rule: The quotient rule is essential for finding the derivative of a fraction. Mastering it is key to differentiating expressions like (1-cos(x))/(1+cos(x)).
  3. Trigonometric Identities: Don't underestimate the power of trigonometric identities! They can help simplify complex expressions and make differentiation much easier. In this case, we used identities for sin(x), cos(x), tan(x/2), and sec(x/2) to get to our final simplified answer.
  4. Step-by-Step Approach: Break down complex problems into smaller, more manageable steps. This makes the problem less daunting and helps you avoid mistakes.

By understanding these concepts and practicing regularly, you'll be able to tackle even the trickiest calculus problems with confidence. Keep up the great work!

Keywords: key takeaways, chain rule, quotient rule, trigonometric identities, step-by-step approach, calculus, composite functions, fractions, simplification, practice

I hope this detailed explanation helped you understand how to find the derivative of y = 2^((1-cos(x))/(1+cos(x))). If you have any more questions, feel free to ask! Happy calculating, guys!