To differentiate \( f(x) = x - \sin(x)\cos(x) \) with respect to \(x\), we'll use standard differentiation rules, such as the product rule and chain rule.

### Step 1: Differentiate \( x \)
The derivative of \( x \) is simply 1:
\[
\frac{d}{dx}(x) = 1
\]

### Step 2: Differentiate \( \sin(x)\cos(x) \)
We need to apply the product rule to differentiate \( \sin(x)\cos(x) \). The product rule states:
\[
\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)
\]
Here, \( u(x) = \sin(x) \) and \( v(x) = \cos(x) \). Let's differentiate each part:
- \( u'(x) = \cos(x) \)
- \( v'(x) = -\sin(x) \)

Now apply the product rule:
\[
\frac{d}{dx}[\sin(x)\cos(x)] = \cos(x)\cos(x) + \sin(x)(-\sin(x)) = \cos^2(x) - \sin^2(x)
\]

### Step 3: Combine the results
Now combine the derivatives from Step 1 and Step 2:
\[
\frac{d}{dx}\left(x - \sin(x)\cos(x)\right) = 1 - (\cos^2(x) - \sin^2(x))
\]
This simplifies to:
\[
\frac{d}{dx}\left(x - \sin(x)\cos(x)\right) = 1 - \cos(2x)
\]
(using the trigonometric identity \( \cos(2x) = \cos^2(x) - \sin^2(x) \)).

### Final Answer:
\[
\frac{d}{dx}\left(x - \sin(x)\cos(x)\right) = 1 - \cos(2x)
\]

Would you like further details or explanations?

### Related Questions:
1. What is the derivative of \( \sin^2(x) + \cos^2(x) \)?
2. How do you apply the chain rule to a composite function like \( \sin(3x) \)?
3. How do you differentiate a product of three functions?
4. What is the second derivative of \( x - \sin(x)\cos(x) \)?
5. How does the derivative change if we consider \( f(x) = x + \sin(x)\cos(x) \)?

### Tip:
Always look out for common trigonometric identities like \( \cos(2x) \), which can simplify your differentiation process.

Learn how to differentiate the function x - sin(x)cos(x) step by step using the product rule and trigonometric identities. This calculus problem is suitable for advanced high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation