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:

The concern that monopolistic competitive firms have about product attribute , services to customer or brand names are aspects of
a. allocative effeciency in the industry
b. collusion in the industry
c. product differentiation
d. concentration ratios

Find dy/dx by implicit differentiation.
1. (2x+3y)^1/3 = x^2
2. x^2y^1/2 = x + 2y^3
3. The demand function for a certain make of ink jet cartridge is
p= -0.01x^2 - 0.1x + 6
Where p is the unit price in dollars and x is the quantity demanded each week, measure in units of a thousand. Compute the elasticity of

Calculate dz/dx using implicit differentiation. given the equation:
(e^(2xy)) + z - (x^2)sec((y)(z^2)) = 2
**note: {the d is the cryllic, or curly d...signifying the partial of z with respect to the partial of x}

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.

Let f(x) = (x^(3x) (4x-1)^8)/sqrt(8x^(2)+10)
If we use the method of logarithmic differentiation to find f '(x), we obtain f '(x) in the form
f ' (x)=[(x^(3x) (4x-1)^8)/sqrt (8x^(2)+10)] [(A(x) + (B(x)/4x-1) + (C(x)/8x^(2)+10)]
where A(x), B(x) and C(x) are functions of x. Find them.
A(x)=______
B(x)=______
C(x)=_

Please see attached for proper formatting.
1. Find the 2nd derivative of g(x)=x^2/(x^2 + 1) .
2. Find d^3/dx^3 (7x^3 - 8x^2 + 2x) .
Use differentials to approximate the following:
1. (127)^(1/3)
2. sin 31°

Differentiation is least effective when it is based on_________.
product quality, service quality, an appeal to the customer's psychological needs, reliability, or physical features
Please advise answer & why. Thanks!