The plain text version simplifies the solution one more time to give Alpha's original answer:
Possible derivation:
d/dx((x^3-3 x^2+2 sqrt(x)-5)/sqrt(x))
Use the product rule, d/dx(u v) = v ( du)/( dx)+u ( dv)/( dx), where u = 1/sqrt(x) and v = x^3-3 x^2+2 sqrt(x)-5:
= (-5+2 sqrt(x)-3 x^2+x^3) (d/dx(1/sqrt(x)))+(d/dx(-5+2 sqrt(x)-3 x^2+x^3))/sqrt(x)
Use the power rule, d/dx(x^n) = n x^(n-1), where n = -1/2: d/dx(1/sqrt(x)) = d/dx(x^(-1/2)) = -1/2 x^(-3/2):
= (d/dx(-5+2 sqrt(x)-3 x^2+x^3))/sqrt(x)+(-5+2 sqrt(x)-3 x^2+x^3) (-1)/(2 x^(3/2))
Differentiate the sum term by term and factor out constants:
= -(-5+2 sqrt(x)-3 x^2+x^3)/(2 x^(3/2))+d/dx(-5)+2 d/dx(sqrt(x))-3 d/dx(x^2)+d/dx(x^3)/sqrt(x)
The derivative of -5 is zero:
= -(-5+2 sqrt(x)-3 x^2+x^3)/(2 x^(3/2))+(2 (d/dx(sqrt(x)))-3 (d/dx(x^2))+d/dx(x^3)+0)/sqrt(x)
Simplify the expression:
= -(-5+2 sqrt(x)-3 x^2+x^3)/(2 x^(3/2))+(2 (d/dx(sqrt(x)))-3 (d/dx(x^2))+d/dx(x^3))/sqrt(x)
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:
= -(-5+2 sqrt(x)-3 x^2+x^3)/(2 x^(3/2))+(-3 (d/dx(x^2))+d/dx(x^3)+2 1/(2 sqrt(x)))/sqrt(x)
Simplify the expression:
= -(-5+2 sqrt(x)-3 x^2+x^3)/(2 x^(3/2))+(1/sqrt(x)-3 (d/dx(x^2))+d/dx(x^3))/sqrt(x)
Use the power rule, d/dx(x^n) = n x^(n-1), where n = 2: d/dx(x^2) = 2 x:
= -(-5+2 sqrt(x)-3 x^2+x^3)/(2 x^(3/2))+(1/sqrt(x)+d/dx(x^3)-3 2 x)/sqrt(x)
Simplify the expression:
= -(-5+2 sqrt(x)-3 x^2+x^3)/(2 x^(3/2))+(1/sqrt(x)-6 x+d/dx(x^3))/sqrt(x)
Use the power rule, d/dx(x^n) = n x^(n-1), where n = 3: d/dx(x^3) = 3 x^2:
= -(-5+2 sqrt(x)-3 x^2+x^3)/(2 x^(3/2))+(1/sqrt(x)-6 x+3 x^2)/sqrt(x)
Simplify the expression:
Answer: |
| = (1/sqrt(x)-6 x+3 x^2)/sqrt(x)-(-5+2 sqrt(x)-3 x^2+x^3)/(2 x^(3/2))