find the derevative (hint try to use limits)
(x^2010 - 1 )\(x -1 )as x approaches1
by definition...
f'(x)= lim (f(x+h)-f(x)/h)
= lim ((x+h)^2010-1/(x+h-1) - (x^2010-1)/(x-1)
Now, expand the first numerator with the binomial expansion, but stop on the second term, because you know h is going to zero.
combine fractions,take h tozero, then x=1, and you get the limit.
what he said
To find the derivative of the function, we can use the limit definition of the derivative. Here's how you can approach the problem step by step:
1. Start with the given function: f(x) = (x^2010 - 1) / (x - 1).
2. Use the limit definition of the derivative, which states that the derivative of a function f(x) at a specific point a is given by the following expression:
f'(a) = lim(h->0) [f(a + h) - f(a)] / h.
3. Substitute a = 1 into the expression:
f'(1) = lim(h->0) [f(1 + h) - f(1)] / h.
4. Plug in the function f(x) = (x^2010 - 1) / (x - 1) into the expression:
f'(1) = lim(h->0) [(1 + h)^2010 - 1] / (1 + h - 1) / h.
5. Simplify the expression inside the limit:
f'(1) = lim(h->0) [(1 + h)^2010 - 1] / h.
6. Expand (1 + h)^2010 using the binomial expansion formula or a calculator:
f'(1) = lim(h->0) [1 + 2010h + (2010(2010-1)h^2)/2 + ... - 1] / h.
7. Cancel out the -1 terms:
f'(1) = lim(h->0) [2010h + (2010(2010-1)h^2)/2 + ...] / h.
8. Divide through by h to simplify:
f'(1) = lim(h->0) [2010 + (2010(2010-1)h)/2 + ...].
9. Take the limit as h approaches 0:
f'(1) = 2010.
Therefore, the derivative of the function (x^2010 - 1) / (x - 1) as x approaches 1 is equal to 2010.