Possible derivation:
d/dx(2 x^(3/2) (sqrt(x) + x^(9/2)))
Factor out constants:
= 2 (d/dx(x^(3/2) (sqrt(x) + x^(9/2))))
Use the product rule, d/dx(u v) = (dv)/(dx) u + (du)/(dx) v, where u = x^(3/2) and v = x^(9/2) + sqrt(x):
= 2 ((sqrt(x) + x^(9/2)) (d/dx(x^(3/2))) + x^(3/2) (d/dx(sqrt(x) + x^(9/2))))
Use the power rule, d/dx(x^n) = n x^(n - 1), where n = 3/2.
d/dx(x^(3/2)) = (3 sqrt(x))/2:
= 2 (x^(3/2) (d/dx(sqrt(x) + x^(9/2))) + (sqrt(x) + x^(9/2)) (3 sqrt(x))/2)
Differentiate the sum term by term:
= 2 (3/2 sqrt(x) (sqrt(x) + x^(9/2)) + (d/dx(sqrt(x)) + d/dx(x^(9/2))) x^(3/2))
Use the power rule, d/dx(x^n) = n x^(n - 1), where n = 1/2.
d/dx(sqrt(x)) = d/dx(x^(1/2)) = x^(-1/2)/2:
= 2 (3/2 sqrt(x) (sqrt(x) + x^(9/2)) + x^(3/2) (d/dx(x^(9/2)) + 1/(2 sqrt(x))))
Use the power rule, d/dx(x^n) = n x^(n - 1), where n = 9/2.
d/dx(x^(9/2)) = (9 x^(7/2))/2:
Answer: = 2 (3/2 sqrt(x) (sqrt(x) + x^(9/2)) + x^(3/2) (1/(2 sqrt(x)) + (9 x^(7/2))/2))