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I'm sorry if I sound too ignorant, I don't have a high level of knowledge in math.

I'm just starting Calc II, yet, I have a tiny understanding of derivatives in higher dimensions.

Having $z(x,y)$, one could treat $y$, for instance, as a constant, in order to get a function with respect to $x$ and then differentiate as one would do in two dimensions. This would, intuitively, give the derivative in the direction of $x$. To my understanding, differentiating with respect to $x$ is different as differentiating with respect to $y$, unless the function $z$ has some sort of symmetry.

But recently I have read in math posts that functions of more than one variable have only one derivative at each point, and that what I have described above is merely the 'directional derivative'.

If this is true, what is this type of derivative they are talking about? What is the equation for it?

Any help/thoughts would be really appreciated.

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    $\begingroup$ There are different kinds of derivatives in higher dimensions. The partial derivative is when you hold all but one variable constant and effectively look at a two-dimensional cross section of the surface; the gradient, which you should read about, takes into account all the variables. In univariate calculus, these are both (except for the vectorial nature of the gradient) the same as a total derivative. $\endgroup$ – gen-z ready to perish Aug 20 '17 at 7:09
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Given a function $f: U \subset \mathbb{R}^n \to V \subset \mathbb{R}^m$ we say $f$ is differentiable at $x \in U$ if there exists a linear map $T: \mathbb{R}^n \to \mathbb{R}^m$ such that,

$$ \lim_{\vec{h} \to \textbf{0}} \frac{\|f(x+\vec{h}) - f(x) - T(\vec{h})\|}{\|\vec{h}\|} = 0 \Rightarrow f(x+\vec{h})-f(x) = T(\vec{h})+\epsilon(x)$$

It follows from the limit definition that when $T$ exists it is unique. Hence $T$ is the best linear approximation to $f$ and we call it the total derivative, derivative, etc and define $T = Df(x)$.

The directional derivative that you are talking about is for functions like $f$ but when $m = 1$. In this case we have $D_{v} f(p) = \nabla f(p) \cdot v=Df(p) \cdot v$ where the LHS is the directional derivative. Although this definition may seem a bit abstract, if our main goal is to study general functions, it would be nice if we had a way to approximate them by simpler functions and if this is possible, it would nice if we already have theory developed for these simpler functions. This is good motivation for the above definition since we have linear algebra.

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