AltDefinition of Derivative is a highly derivative exploration of what the derivative of sqrt[x] is. It owes much to Paul Foerster, whose Explorations in Calculus book is a prized possession of mine. You can do the same for cube root of x, or x to the 4th power. Foerster’s original did the same process with x to the 5th power.
put sqrt of x into Limit as x->c of [f(x) – f(c)]/(x-c). Write it out. You can factor the denominator (if it is a square root or cube root), or factor the numerator (if it is a positive natural number exponent), and cancel factors. Then substitute c for x, and voila! the derivative at the point x=c.
The factoring pattern is x^3 – c^3 = (x – c) (x^2 + xc + c^2)
or x^4 – c^4 = (x-c)(x^3 + x^2c + xc^2 + c^3)
or (x – c) = (x^.5 – c^.5) (x^.5 + c^.5)