What are the first and second derivatives of y= x^(2/3) * (2.5 - x)
First way: Just multiply it out:
y = 5/2 x^(2/3) - x^(5/3)
y' = 5/3 x^(-1/3) - 5/3 x^(2/3)
y'' = -5/9 x^(-4/3) - 10/9 x^(-1/3)
Second way: Use product rule:
y = x^(2/3) * (5/2 - x)
y' = 2/3 * x^(-1/3) * (5/2 - x) + x^(2/3) * -1 = 5/3 x^(-1/3) * (1-x)
y'' = 2/3 * (-1/3) x^(-4/3) * (5/2 - x) + 2/3 * x^(-1/3) * -1 - 2/3 * (x^-1/3)
= -5/9 x^(-4/3) * (2x+1)
collect terms and you'll find they agree.