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)
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