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

1 Answer
Feb 6, 2016

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)

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here g(x) = (x+2)^2

and usingcolor(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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