# Proof differentiation of functions

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)

https://brainmass.com/math/derivatives/proof-differentiation-functions-477912

#### 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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