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

1 Answer
Feb 6, 2016

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)#
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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 #
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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)#

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