How would you determine the unsimplified derivatives of these two questions?

Question

a) y= (3x^2-x)(2x^3-3)

———————

2x-5

b) y= (x^2-1)^2(x^3-5)^2

———————

-2x^2+1

i believe I would use the power/product/chain rule combo however I’m unsure of how to begin

Answer 1

Hi,

y= (3x^2-x)(2x^3-3)

———————

2x-5

your mean is y= (3x^2-x)(2x^3-3)/ (2x-5)

If yes, Please follow me

1) first this function follows division rule that top is f(x) =(3x^2-x)(2x^3-3) and bottom is g(x)= (2x-5)

you know the rule is y' = (f' g - g' f ) / g²

f' = (6x - 1) ( 2x^3-3) + ( 6x²) (3x^2-x) = 30 x⁴ - 8 x³- 18x + 3

g' = 2

Now plug in to the the y' = (f' g - g' f ) / g² = [(30 x⁴ - 8 x³- 18x + 3)(2x-5) - 2(3x^2-x)(2x^3-3)] / (2x-5) ²

---------------------------------------------------------

y= (x^2-1)^2(x^3-5)^2

———————

-2x^2+1

your mean is y = [(x^2-1)^2(x^3-5)^2] / ( -2x^2+1)

one more time follow division rule this time top is f(x) = [(x^2-1)^2(x^3-5)^2] and bottom is g(x) = -2x^2+1

I think only you need to know the f'(x) then follow the rule such as what i said on above!

f'(x) = 2 ( x^2-1) ( 2x) ( (x^3-5)^2 + 2 (x^3-5) ( 3x²)(x^2-1)^2

g'(x) = -4x

Plug in y' = (f' g - g' f ) / g²

I hope it is useful,

Minoo