How do you differentiate #f(x)=csc5x * csc2x# using the product rule?

Question

How do you differentiate #f(x)=csc5x  *  csc2x# using the product rule?

Ezekiel Alexander · Accepted Answer

#f=csc5x, g=csc2x#
#f'=-csc5xcot5x*5, g'=-csc2xcot2x*2#
#f'(x)=fg'+gf'=(csc5x)(-2csc2xcot2x)+(csc2x)(-5csc5xcot5x)# Apply the product rule to find the derivative of f(x), but first divide the products into f and g. Then, find the derivative of each function independently and combine all of the results into the product rule.

Luke Alvarez · Answer

undefined To differentiate $ f(x) = \csc(5x) \cdot \csc(2x) $ using the product rule, you can follow these steps:

1. Recall the product rule: $ (uv)' = u'v + uv' $.
2. Let $ u = \csc(5x) $ and $ v = \csc(2x) $.
3. Find the derivatives of $ u $ and $ v $:
   - $ u' = -5\csc(5x) \cot(5x) $
   - $ v' = -2\csc(2x) \cot(2x) $.
4. Apply the product rule: 
   - $ f'(x) = u'v + uv' $.
5. Substitute the derivatives and original functions into the formula:
   - $ f'(x) = (-5\csc(5x) \cot(5x)) \cdot \csc(2x) + \csc(5x) \cdot (-2\csc(2x) \cot(2x)) $.
6. Simplify the expression if needed.