CALCULUS PROBLEMS:HOW TO DIFFERENTIATE PRODUCT FUNCTIONS

Differentiate (x-1)^2 (x+2)^3

  

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 Best way is to first multiply it out" 

"product rule, however differentiate each function using the chain rule " 

"Use the product rule" 

Well i say take the ln (log to the base of e (natural log)) 

that'd be the best way. 

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Y = (x-1)^2 (x+2)^2 

lnY = ln [ (x-1)^2 (x+2)^3]

Now use these formulas to simplify, 

log_a b + log_a c = log_a (bxc) 

log_a b - log_a c = log_a (b-c) 

log_a b^c = clog_a b 

here log_a means log to the base of a 

to our question a = e; so log_a = log_e
log_e = ln

lnY = ln(x-1)^2 + ln(x+2)^3 

lnY = 2ln(x-1) + 3ln(x+2) <= see how small all that became

Now differentiante 

(1/y)dy/dx = 2/(x-1) + 3(x+2)

dy/dx = y [ 2/(x-1) + 3/(x+2) 

dy/dx = y [2(x+2) + 3(x-1)] / (x-1)(x+3) 

dy/dx = (x-1)^2 (x+2)^3 [ 2(x+2) + 3(x-1) ] / (x-1)(x+2) 

dy/dx = (x-1) (x+2)^2 [ 2(x+2) + 3(x-1)] 

dy/dx = (x-1) (x+2)^2 (5x +1) 

dy/dx = (5x+1)(x-1) (x+2)^2.

OR

 Using Product Rule , f ` (x) is given by :- 

Udv/dx +Vdu/dx

2(x - 1) (x + 2) ³ + 3(x + 2) ² (x - 1) ²

(x - 1) (x + 2) ² [ 2 (x + 2) + 3 (x - 1) ]

(x - 1) (x + 2) ² [ 5x + 1 ] 

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