Proving differentiability for a function from the definition

member 731016
Homework Statement
Please see below
Relevant Equations
Please see below
For this problem,
1715475811269.png

The solution is,
1715475847832.png

However, does someone please know why we allowed to assume that the derivative exists for f i.e ##f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}##?

Thanks!
 
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ChiralSuperfields said:
Homework Statement: Please see below
Relevant Equations: Please see below

For this problem,
View attachment 345049
The solution is,
View attachment 345050
However, does someone please know why we allowed to assume that the derivative exists for f i.e ##f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}##?

Thanks!
Because of the first sentence: Let ##f## be a differentiable function.
 
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There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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