Prove that if f(x) = x^alpha, where alpha = 1/n for some n in N (the natural numbers), then y = f(x) is differentiable and f'(x) = alpha x^(alpha - 1). Progress I have made so far: I have managed to prove, (x^n)' = n x^(n - 1) for n in N and x in R both from the definition of differentiation involving the limit and the binomial theorem or equivalently using induction on n. Feel free to use this result although anything else should be made rigorous. It should be possible to prove this by the basic definition of the derivative. Thanks! Please no fancy inversefunction theorem, just the basic definition of the derivative...

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To show that f(x) is differentiable at x, you should prove that this limit exists:
lim[f(x+h)-f(x)]/h ; (h->0)
and then the result of this limit is f'(x). Therefore, we can narrow down the problem to finding this limit:

f'(x)=lim[(x+h)^(1/n)-x^(1/n)]/h ; (h->0)

Now, I show you how to find the limit, but before starting, observe that we will actually face 0/0 case:

This solution shows how to solve for various calculus problems, including differentiation of functions using the product rule, the quotient rule, and the chain rule, as well as how to calculate integrals.

A specialty store that cater to the needs of teenagers by offering designer clothing in a trendy atmosphere is most likely pursuing which type of strategy?
a. differentiation advantage
b. learning curve advantage
c. cost advantage
d. Differentiation combined with cost leadership

Horizonal integration supports the achievement of a ______________ strategy?
(a) cost leadership
(b) differentiation
(c) focus
(d) cost leadership and differentiation
(e) stuck in the middle

THE DIFFERENTIATION OF ALGEBRAIC FUNCTIONS
Use implicit differentiation to find the dy/dx:
3xy + x2y2 = 1
xy1/2 - 2x + y = 8
(x + y)3 = x3 + y3
3y2 + 5x2 -2x = 5
y4 = x3y2 + x2y3 - 3
Find the indicated higher order derivative of the following functions:
1. Find the 2nd derivative of f(x) = 7x^5 - 4

Consider the curve xy^2 + y = 3 - x^2. Use implicit differentiation to find the derivative dy/dx
Hence determine the tangent to the curve at the point (-2,1)
A stone dropped into a still pond sends out a circular ripple whose radius increases at a constant rate of 0.5ms-1. How rapidly is the area enclosed by the ripple inc