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# Proof differentiation of functions

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1) Let n be a natural number, and let f: R-->R be the function f(x) :=x^n. Show that f is differentiable on R and f '(x)=nx^n-1 for all x belonging to R
(Use induction)

2) Let n be a negative integer, and let f:R - {0} -->R be the function f(x):=x^n, show that f is differentiable on R and f '(x)=nx^n-1 for all x belonging to R - {0}
(Use problem 1)

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#### Solution Preview

We have:
(1.1)
Where n is a positive integer
The derivative is defined as
(1.2)
For we have:
(1.3)
For we have:

(1.4)
We assume that
(1.5)
And we show by induction that if this is true then
(1.6)

We ...

#### Solution Summary

The expert examines proof differentiation of functions.

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