Question

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

Answer 1

`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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