Definition of a Derivative/Limit

[cse63146]

I'm in a multi-variable calculus course, and we need to know definitions for the test.

Is this a correct definition of a derivative:

Let $$f:R^n \rightarrow R^p$$ f is diffrentiable at x iff there exists a matrix (p x n) Df(x) such that $$lim (h\rightarrow0)\frac{f(x - h) -f(x) -Df(x)h}{||h||} = 0$$

My professor gave us quite a few definitions for derivative, some involve gradients and epsilon. Just wondering if the above also works.

and is this a correct definition of a limit:

$$lim (x \rightarrow x_0) F(x) = A$$ there exists $$\epsilon > 0$$ such that $$\delta>0 \Rightarrow$$ $$0<||x - x_0|| <\delta \Rightarrow ||f(x) - A|| < \epsilon$$

 

[mighty2000]

Yes, both of these definitions are correct for a derivative and a limit. The first definition for a derivative is known as the "matrix definition" and is commonly used in multivariable calculus. The second definition for a limit is known as the "epsilon-delta definition" and is commonly used in calculus. Both definitions capture the essential properties of a derivative and a limit, respectively. However, there may be other equivalent definitions that your professor has given you that may also be correct. It's important to understand the different definitions and how they relate to each other in order to fully grasp the concept of derivatives and limits. Good luck with your course!