How do you differentiate #f(x)=(x+2)^2(x-5)^3# using the product rule?
1 Answer
# f'(x) = (x-5)^2(x+2)(5x - 4) #
Explanation:
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)("----------------------------------------------------")#