How do you differentiate #f(x)=(1/(3x))*x*e^x-x*cosx# using the product rule?

Question

Emily Ammerman · Accepted Answer

Please see below.

Simplify first #f(x) = 1/3e^x-xcosx# We do not need the product rule for the first term, although you may use it if you like. #d/dx(1/3e^x) = d/dx(1/3)e^x+1/3 d/dx(e^x) = 0e^x + 1/3e^x = 1/3e^x# For the second term, we need the product rule. (unless we do something unusual.) #d/dx(xcosx) = d/dx(x) * cosx + x * d/dx(cosx)# # = cosx-xsinx# Putting it all together, #f'(x) = 1/3e^x + cosx - xsinx#

Adam Adkins · Answer

undefined To differentiate the function $ f(x) = \frac{1}{3x} \cdot x \cdot e^x - x \cdot \cos(x) $ using the product rule, follow these steps:

1. Identify the two functions being multiplied together: $ u(x) = \frac{1}{3x} \cdot x \cdot e^x $ and $ v(x) = -x \cdot \cos(x) $.
2. Apply the product rule: $ (uv)' = u'v + uv' $.
3. Compute the derivatives of $ u(x) $ and $ v(x) $:
   - $ u'(x) = \left(\frac{1}{3x} \cdot x \cdot e^x\right)' $
   - $ v'(x) = (-x \cdot \cos(x))' $.
4. Calculate the derivatives using the product rule and the chain rule where necessary.
5. Add the two results together to get the derivative of the original function $ f(x) $.