If `f(x)=xcosx`, find `f'(x)`.

Explanation

To find the derivative of the function \( f(x) = x \cos x \), we will use the product rule of differentiation. The product rule states that if you have two functions multiplied together, say \( u(x) \) and \( v(x) \), then the derivative is given by:

\[ f'(x) = u'v + uv' \]

where \( u' \) is the derivative of \( u \) and \( v' \) is the derivative of \( v \).

### Step 1: Identify the functions
In our case, we can identify:
- \( u = x \)
- \( v = \cos x \)

### Step 2: Differentiate each function
Now, we need to find the derivatives of \( u \) and \( v \):
- The derivative of \( u \) with respect to \( x \) is:
\[ u' = \frac{d}{dx}(x) = 1 \]
- The derivative of \( v \) with respect to \( x \) is:
\[ v' = \frac{d}{dx}(\cos x) = -\sin x \]

### Step 3: Apply the product rule
Now we can apply the product rule:
\[ f'(x) = u'v + uv' \]
Substituting the values we found:
\[ f'(x) = (1)(\cos x) + (x)(-\sin x) \]
This simplifies to:
\[ f'(x) = \cos x - x \sin x \]

### Final Answer
Thus, the derivative of the function \( f(x) = x \cos x \) is:
\[ f'(x) = \cos x - x \sin x \]

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