Question

I have been asked to solve this two different ways. The first way is to use the chain rule and then simplify (which I have already done properly), and the second way is to simplify and then differentiate (not necessarily with the chain rule). I am stuck on how to simplify this question and then how to differentiate it.

Square Root(9x^2) ??

Also I am stuck on how to use the chain rule to differentiate and then simplify this question, as well as the second way where I have to simplify and then differentiate it:

(8x^3)^2/3

I would appreciate if someone would be able to explain these two questions to me, thank you.

Answer 1

y = [ 9 x^2 ] ^.5 ??

let z = 9 x^2 then dz/dx = 18 x

then we have y = z^.5

dy/dz = .5 z^-.5 = .5/3x

but chain rule
dy/dx = dy/dz*dz/dx (.5 /3x)(18 x)
= 9/3 = 3 at last

simplify first
y = (9 x^2)^.5 = 3x
dy/dx = 3 the end

Answer 2

(8x^3)^2/3

I am not going to struggle through that with the chain rule. Go about it the same way as I did upstairs, z = 8 x^3

simplify method:
remember (8x^3)^(1/3) = 2x
(2x)^2 = 4 x^2
y = 4x^2
dy/dx = 8 x the end