Help with derivitives [calculus]?

Question

Well this practice problem in my AP Calculus practice book has a strange problem giving me a hard time. It says:
"Each limit represents the derivate of some function f at some number a. State such an f and a in each case".

and the problem is...
lim h---> o of...
(1+h)^10 - 1
---------------- [divided by]
h

I tried using the definition of a derivative and also the alternate definition of a derivative [aka: derivative at a point] and I didn't quite get the answer from the answer key. So, how is this actually done?

Answer 1

Consider the function f(x) = x^10. For f'(1) we would take lim(h -> 0) [f(1+h) - f(1)]/h = lim[(1+h)^10 - 1]/h.

Answer 2

you may desire to coach the chain rule. This function has 5 nested applications: cos, sqrt, sin, tan, and a linear function (?x). keep in mind that cos' = -sin, sqrt' = a million / (2*sqrt), sin' = cos, tan' = sec^2 or a million + tan^2, and (ax)' = a y = cos?(sin(tan?x)) y' = -sin?(sin(tan?x)) * (?(sin(tan?x)))' = -sin?(sin(tan?x)) * (a million / 2?(sin(tan?x))) * (sin(tan?x))' = -sin?(sin(tan?x)) * (a million / 2?(sin(tan?x))) * cos(tan?x) * (tan?x)' = -sin?(sin(tan?x)) * (a million / 2?(sin(tan?x))) * cos(tan?x) * (a million + tan^2(?x)) * (?x)' = -sin?(sin(tan?x)) * (a million / 2?(sin(tan?x))) * cos(tan?x) * (a million + tan^2(?x)) * ? I bypass away any needed simplication to you, yet I do see a ?(sin(tan?x)) in the two the numerator and denominator.