How do you differentiate #f(x)=(x+2)^2(x-5)^3# using the product rule?

Question

Cameron Alexander · Accepted Answer

# f'(x) = (x-5)^2(x+2)(5x - 4) #

for a function f)x) = g(x).h(x) ie. a product of 2 functions then f'(x) = g(x).h'(x) + h(x).g'(x).............................(A) #color(black)("-------------------------------------") # here g(x) # = (x+2)^2 # and using#color(blue)(" chain rule ")# g'(x) # = 2(x+2) d/dx (x+2) =2(x+2) .1 = 2(x+2)# #color(black)("-------------------------------------------")# and h(x) # = (x-5)^3# #color(blue)(" again using chain rule ")# h'(x)# = 3(x-5)^2 d/dx (x-5) = 3(x-5)^2 .1 = 3(x-5)^2 # #color(black)("--------------------------------------------------")# substituting back into ( A) gives : f'(x) # = (x+2)^2 . 3(x-5)^2 + (x-5)^3 .2(x+2) # # = 3(x+2)^2(x-5)^2 + (x-5)^3. 2(x+2)# take out common factors # (x+2)(x-5)^2# # = (x-5)^2(x+2)[3(x+2) + 2(x-5)] # #rArr f'(x) = (x-5)^2(x+2)[3x+6+2x-10] = (x-5)^2(x+2)(5x-4)# #color(black)("----------------------------------------------------")#

Ezekiel Alexander · Answer

undefined To differentiate $ f(x) = (x+2)^2(x-5)^3 $ using the product rule, follow these steps:

1. Identify the two functions being multiplied: $ u(x) = (x+2)^2 $ and $ v(x) = (x-5)^3 $.
2. Apply the product rule: $ f'(x) = u'(x)v(x) + u(x)v'(x) $.
3. Find the derivatives of $ u(x) $ and $ v(x) $:
   - $ u'(x) = 2(x+2)(1) $ (using the power rule and chain rule)
   - $ v'(x) = 3(x-5)^2 $ (using the power rule and chain rule)
4. Substitute the derivatives into the product rule formula: $ f'(x) = 2(x+2)(x-5)^3 + (x+2)^2(3(x-5)^2) $.
5. Simplify the expression if necessary.