10 / 2/3\ \1 - 5*x /
(1 - 5*x^(2/3))^10
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:
Now simplify:
The answer is:
9 / 2/3\ -100*\1 - 5*x / ------------------ 3 ___ 3*\/ x
8 / 2/3\ / 2/3\ | -1 + 5*x | 100*\-1 + 5*x / *|10 - -----------| | 2/3 | \ 9*x / ------------------------------------- 2/3 x
/ 2\ 7 | / 2/3\ / 2/3\ | / 2/3\ |400 5*\-1 + 5*x / 2*\-1 + 5*x / | 200*\-1 + 5*x / *|--- - --------------- + ----------------| |3*x 5/3 7/3 | \ x 27*x /