Derivatives?

Question

find the derivative of ((y^2)/(y+6))^4

Answer 1

Let p = [y^2/(y+6)]^4

Differentiating on both sides with respect to y

dp/dy = 4 [((y+6)(2y) - y^2)/(y+6)^2]^3
dp/dy = 4 [(2y^2 + 12y - y^2)/(y+6)^2]^3
dp/dy = 4[(y^2 + 12y)/(y + 6)^2]^3

I hope you can do the rest of the simplification... as its extremely difficult to type all that...

Answer 2

This is just a straightforward use of the chain rule and then the quotient rule. It becomes
4(original)^3*(derivative of what's inside)
using the chain rule. Then use the quotient rule for second bracket.

Answer 3

f ` (y) = [(y + 6)^(4).2y - y².4(y+6)³] / (y + 6)^8
f `(y) = 2y.(y + 6)³ [ y + 6 - 2y] / (y + 6)^(8)
f ` (y) = 2y.(6 - y) / (y + 6)^(5)

Answer 4

y=(u)^n ==> y'=n*(u)^(n-1)*u'
y=u/p ==>y'=(u'p-p'u)/p^2
g(y)=y^2 ==> g'(y)=2y
f(y)=y+6 ==>f'(y)=1
h(y)=g(y)/f(y) ==> h'(y)=(2y(y+6)-(y^2))/(y+6)^2
the answer is: 4((y^2)/(y+6))^3 *(2y(y+6)-(y^2))/(y+6)^2