The discussion focuses on deriving the cube root formula using Newton-Raphson's method, specifically the equation x - y³ = 0. The established solution is (2y + (x/y²))/3, which simplifies the process of finding cube roots. Participants clarify that implicit differentiation is unnecessary for this derivation; instead, one should solve f(x) = x³ - n for its roots using the Newton-Raphson iteration formula xn+1 = xn - f(xn)/f'(xn). This approach emphasizes the importance of correctly identifying variables to avoid confusion.
PREREQUISITESStudents in calculus, mathematicians interested in numerical methods, and anyone seeking to understand the derivation of cube roots using iterative techniques.
DAPOS said:Homework Statement
Derive cube root formula using Newton-Rhapson's method. x - y^3 = 0.
Homework Equations
xn + 1 = xn - f(xn)/f'(xn)
The Attempt at a Solution
I know that the solution is (2y + (x/y^2))/3
I tried using implicit differentiation and stuff but I can't get this out. Any tips?
LCKurtz said:I think your use of x and y is confusing you. If you want the cube root of ##n## you might solve ##f(x) = x^3 - n## for its roots using your above formula. No implicit differentiation needed and you can switch the names of the variables at the end if you want to.