Finding The Derivative Of F(x) = Cos(cos(x))

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Hey guys! Let's dive into a fun little calculus problem. We're gonna find the derivative of the function f(x) = cos(cos(x)). Sounds a bit tricky, right? Don't sweat it, we'll break it down step by step and make it super clear. This is the kind of problem that often pops up in math exams, so understanding it is a total win. We will go through the process of calculating the derivative, and at the end we can select the correct option. Ready to roll?

Understanding the Problem and the Chain Rule

Alright, so the core of this problem is understanding the chain rule. Basically, the chain rule is our secret weapon when we have a function within a function – a composite function. In our case, we've got the cosine function on the outside and another cosine function lurking on the inside. The chain rule states: if you have a function f(g(x)), then its derivative is f'(g(x)) * g'(x). In simpler terms, you find the derivative of the outer function, keeping the inner function as is, and then you multiply that by the derivative of the inner function. Easy peasy, right?

Let's break down the given function f(x) = cos(cos(x)). Here, our outer function is cos(u), and our inner function is g(x) = cos(x), where u = cos(x). To solve this, we will implement the chain rule in a step-by-step fashion. First, we need to know the derivatives of both the cosine function and its inner component. The derivative of cos(x) is -sin(x). We'll need this later, so keep it in mind. This is a fundamental fact, and if you haven't memorized it, now is a good time to do so! The chain rule helps us deal with the composition of functions. In this case, we have a cosine function containing another cosine function. We must first deal with the outer function, and then consider the inner function.

So, as a preliminary, let's establish the derivative of the outer function. This derivative will be evaluated at cos(x), the inner function. Then, we will have to multiply it by the derivative of the inner function to arrive at our answer. Remember, the derivative of cos(x) is -sin(x). Now, let’s get to the real work! By applying the chain rule, we can begin to find the answer. The derivative of the outer function, cos(u) where u=cos(x) is -sin(u), or, -sin(cos(x)).

Now, we multiply by the derivative of the inner function, which is cos(x). The derivative of cos(x) is -sin(x). Thus, we will multiply -sin(cos(x)) by -sin(x). The product of these two expressions is sin(cos(x))sin(x).

Step-by-Step Solution

  1. Identify the outer and inner functions: Our outer function is cos(u) and the inner function is u = cos(x).
  2. Find the derivative of the outer function: The derivative of cos(u) is -sin(u). Since u = cos(x), this becomes -sin(cos(x)). This is the result from differentiating the outer component of the equation.
  3. Find the derivative of the inner function: The derivative of cos(x) is -sin(x). This is the result from differentiating the inner component of the equation.
  4. Apply the chain rule: Multiply the derivative of the outer function by the derivative of the inner function. So, we have: -sin(cos(x)) * (-sin(x)) = sin(cos(x))sin(x). Thus, the correct answer to this question must be sin(cos(x))sin(x).

Matching the Answer

Now, let's match our derived answer with the given options:

  • a. f'(x) = 2cos(x) - Nope. This looks nothing like our answer.
  • b. f'(x) = -sin(cos(x))cos(x) - Close, but not quite. Notice how it is very similar to what we arrived at earlier, when we only dealt with the derivative of the outer function. This is not the correct answer, since we did not complete all the steps.
  • c. f'(x) = 2cos(x)sin(x) - Nah, this one's completely off the mark.
  • d. f'(x) = sin(cos(x))sin(x) - Bingo! This matches exactly what we calculated.
  • e. f'(x) = sin(sin(x)) - Nope. Not even close.

So, the correct answer is d. f'(x) = sin(cos(x))sin(x). We got there by correctly applying the chain rule, which is a key concept in calculus. It's all about taking things step-by-step and keeping track of the different functions involved.

Conclusion: Mastering the Chain Rule

Alright, guys, that's a wrap! We've successfully navigated the derivative of f(x) = cos(cos(x)). We've highlighted the importance of the chain rule. This method is incredibly useful for solving a wide range of calculus problems, especially those involving composite functions. Remember, the chain rule is the magic trick to differentiate a function within a function. First, find the derivative of the outer function, leaving the inner function untouched. Then, multiply that result by the derivative of the inner function. Practice these steps, and you will be acing these problems in no time. The key is to break down the problem and apply the chain rule methodically. The chain rule is one of the most important rules of differentiation. Make sure that you are confident with it.

Understanding derivatives is fundamental to grasping more advanced calculus concepts. Keep practicing, and you'll find that these types of problems become easier and more intuitive. Now, go forth and conquer those derivatives! Keep in mind that math can be fun! If you are ever struggling with a problem, don't be afraid to break it down into smaller steps. This is a powerful technique, and you can apply it to a variety of mathematical concepts. Remember to always double-check your work, and always ask questions. You can also work together with your classmates and your teacher. This can help you to understand more effectively. Good luck, and keep up the great work! That's all for today, thanks for tuning in!