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**Directional derivative and "non-appropriate" solution?**

## Homework Statement

Given following normal vectors [tex]u_1,__2,...,u_n[/tex] which are basis for [tex]R^N[/tex]

Prove that there are n scalars which satisfy:

[tex]f_x(a)=\sum_{i=1}^na_if_{u_i}(a)[/tex], given that f is differentiable at [tex]a\in R^n[/tex]

## Homework Equations

I know that if f is differentiable at [tex]a\in R^n[/tex] for every [tex]t=(t_1,t_2,...,t_n)[/tex] happens [tex]f_u=\sum_{i=1}^n f_{x_i}(a)t_i[/tex]

## The Attempt at a Solution

I changed from the standard basis [and it's cartesian coordinate system] to the new basis [tex]B=\{u_1,u_2,...,u_n\}[/tex]

because B is basis i can write [x] as [tex]k_1u_1+k_2u_2+...+k_nu_n[/tex]

and then:

[tex]f_x([a]_e)=f_{k_1u_1+k_2u_2+...+k_nu_n}([a]_B)=\sum_{i=1}^n f_{u_i}([a]_B)k_i[/tex]

And the proof is finished.

What is bothers me is the fact that the "relevant equation" was proved using the standard basis, so I'm not sure that my solution is appropriate given how easy it becomes with such a trick.

Do you think my proof is valid?

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