Use the quotient rule to find the derivative of the following function.𝑓(𝑥)=cos(𝑥)ln(𝑥)𝑓′(𝑥)=

To find the derivative of the function $f(x) = \cos(x) \ln(x)$ using the product rule, we break down the problem as follows:

1. Break Down the Problem

Identify the two functions involved:

• $u = \cos(x)$
• $v = \ln(x)$

2. Relevant Concepts

The product rule states that the derivative of a product of two functions is given by: $(fg)' = f'g + fg'$ Thus, we need to find $u'$ and $v'$.

3. Analysis and Detail

1. Compute $u'$:

   • $u = \cos(x)$
   • Therefore, $u' = -\sin(x)$

2. Compute $v'$:

   • $v = \ln(x)$
   • Thus, $v' = \frac{1}{x}$

3. Apply the product rule: $f'(x) = u'v + uv' = (-\sin(x)) \ln(x) + \cos(x) \left(\frac{1}{x}\right)$

4. Verify and Summarize

Combining the two terms gives us the final expression for the derivative: $f'(x) = -\sin(x) \ln(x) + \frac{\cos(x)}{x}$

Final Answer

$f'(x) = -\sin(x) \ln(x) + \frac{\cos(x)}{x}$