____________ 3 / 2 \/ x*(6 - x)
(x*(6 - x)^2)^(1/3)
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
Apply the product rule:
; to find :
Apply the power rule: goes to
; to find :
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
Differentiate term by term:
The derivative of the constant is zero.
The derivative of a constant times a function is the constant times the derivative of the function.
Apply the power rule: goes to
So, the result is:
The result is:
The result of the chain rule is:
The result is:
The result of the chain rule is:
Now simplify:
The answer is:
/ 2 \ 3 ___ 2/3 |(6 - x) x*(-12 + 2*x)| \/ x *|-6 + x| *|-------- + -------------| \ 3 3 / -------------------------------------------- 2 x*(6 - x)
/ 2/3 3 ___ \ ||-6 + x| 2*\/ x *sign(-6 + x)| (-2 + x)*|----------- + --------------------| 2/3 | 2/3 3 __________ | 2/3 2/3 |-6 + x| *(-2 + x) \ x \/ |-6 + x| / 2*|-6 + x| *(-2 + x) 2*|-6 + x| *(-4 + x) - -------------------- + --------------------------------------------- - ---------------------- + ---------------------- 5/3 3*x 2/3 2/3 x x *(-6 + x) x *(-6 + x) ------------------------------------------------------------------------------------------------------------------------ -6 + x
/ 2/3 3 ___ \ / 2/3 3 ___ 2 3 ___ \ / 2/3 3 ___ \ / 2/3 3 ___ \ ||-6 + x| 2*\/ x *sign(-6 + x)| | |-6 + x| \/ x *sign (-6 + x) 2*sign(-6 + x) 6*\/ x *DiracDelta(-6 + x)| ||-6 + x| 2*\/ x *sign(-6 + x)| ||-6 + x| 2*\/ x *sign(-6 + x)| (-2 + x)*|----------- + --------------------| 2*(-2 + x)*|- ----------- - ------------------- + ----------------- + --------------------------| 2*(-2 + x)*|----------- + --------------------| 2*(-4 + x)*|----------- + --------------------| 2/3 | 2/3 3 __________ | | 5/3 4/3 2/3 3 __________ 3 __________ | 2/3 2/3 2/3 2/3 | 2/3 3 __________ | | 2/3 3 __________ | 2/3 2*|-6 + x| \ x \/ |-6 + x| / \ x |-6 + x| x *\/ |-6 + x| \/ |-6 + x| / 5*|-6 + x| *(-2 + x) 8*|-6 + x| *(-4 + x) 6*|-6 + x| *(-2 + x) 10*|-6 + x| *(-4 + x) \ x \/ |-6 + x| / 2*(-2 + x)*sign(-6 + x) \ x \/ |-6 + x| / 10*|-6 + x| *(-2 + x) 4*(-2 + x)*sign(-6 + x) 4*(-4 + x)*sign(-6 + x) ------------- - --------------------------------------------- + ------------------------------------------------------------------------------------------------- + ---------------------- - ---------------------- + ---------------------- - ----------------------- - ----------------------------------------------- - ----------------------- + ----------------------------------------------- + ----------------------- - ---------------------------- + ---------------------------- 2/3 2 9*x 8/3 2/3 2 2/3 2 5/3 3*x*(-6 + x) 5/3 3 __________ 3*x*(-6 + x) 5/3 2/3 3 __________ 2/3 3 __________ x *(-6 + x) 3*x 3*x x *(-6 + x) x *(-6 + x) 3*x *(-6 + x) 3*x *\/ |-6 + x| 3*x *(-6 + x) 3*x *(-6 + x)*\/ |-6 + x| 3*x *(-6 + x)*\/ |-6 + x| -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -6 + x