calculusderivativesgraphing-functions
I know how the derivative of y with respect to x ($\frac{dy}{dx}$) represents the slope of the the line tangent to the curve at the point with x-coordinate as 'x', in the x-y plane ; this is when we speak of 2D space.
What I wish to know is what these derivatives represent in higher dimensions.
Eg. in 3D, what does the value of $f'(x,y)$ represent?
Take in mind I am still in high school so explain in a reasonably understandable language.
Thanks..
The second derivative tells you something about how the graph curves on an interval.
If the second derivative is always positive on an interval $(a,b)$ then any chord connecting two points of the graph on that interval will lie above the graph. If the second derivative is always negative on the interval $(a,b)$ then any chord connecting two points of the graph on that interval will lie below the graph.
In the graph below of $y=x(x-1)(x+1)$ the graph has a negative second derivative on the interval $(-\infty,0)$ and a positive second derivative on the interval $(0,\infty)$ so it is concave down and concave up, respectively on the two intervals.
Another way of expressing the same idea is that if a continuous second differentiable function has a positive second derivative at point $(x_0,y_0)$ then on some neighborhood of $(x_0,y_0)$ the tangent line at $(x_0,y_0)$ lies below the graph (except at the point of tangency). If the second derivative is negative at the point of tangency the tangent line lies above the graph on some neighborhood of the point of tangency (except at the point of tangency).
The concept generalizes in a completely straightforward way, with the largest difference mostly being that you can no longer draw the higher dimensional pictures. Let's formalize the geometric tangent plane as follows.
Let $f:\mathbb{R}^m \rightarrow \mathbb{R}^n$ be a differentiable function. We consider the graph $\Gamma_f$ of the function as the set of points $(\mathbf{x},f(\mathbf{x})) \in \mathbb{R}^m \times \mathbb{R}^n$. The tangent plane to a point $(\mathbf{x}_0,f(\mathbf{x}_0)) \in \Gamma_f$ is then defined by the set of points $(\mathbf{x},D_{\mathbf{x}_0}f(\mathbf{x}-\mathbf{x}_0) + f(\mathbf{x}_0))$, where $D_{\mathbf{x}_0}f:\mathbb{R}^m \rightarrow \mathbb{R}^n$ is the derivative of $f$ at $\mathbf{x}_0$.
For example, given a function $f:\mathbb{R}\rightarrow \mathbb{R}$, this reduces to your regular notion of the tangent line given by $$y = \frac{df}{dx}(x-x_0) + f(x_0).$$ In general, given a function $f:\mathbb{R}^m \rightarrow \mathbb{R}^n$, the tangent plane will be an $m$-dimensional plane which sits within the ambient space $\mathbb{R}^m \times \mathbb{R}^n$. Given a function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ for example, the tangent plane is a $2$-dimensional plane defined within $\mathbb{R}^2\times \mathbb{R} \cong \mathbb{R}^3$, as you can see through explicit drawings.
All of this is really saying that the derivative is the linear mapping which best approximates a given function in some local neighborhood. The derivative matrix of $f:\mathbb{R}^m \rightarrow \mathbb{R}^n$ is an $m \times n$ matrix whose $(i,j)$th component tells you how much the tangent plane is inclined in the $j$th direction of the codomain space $\mathbb{R}^n$ when moving in the $i$th direction of the domain space $\mathbb{R}^m$.
Let me finally note that this geometric view of tangent planes is very useful for intuition, but often not so useful in practice. I've been careful to call the objects I've defined tangent planes, because tangent spaces are refer to another concept with precise meaning in differential geometry. Tangent spaces are abstract formalizations of linearizations of smooth manifolds, and derivatives are really linear maps between the tangent spaces of respective manifolds. I think Spivak's book introduces the concept of tangent spaces in the last few chapters.
Best Answer
I would say prime notation is a bit out of its depth here. If you are taking the derivative of a multivariate function $f(x,y)$, you would need to specify the variable with respect to which you are differentiating.
For example, letting $f(x,y)=x+y$: $$\frac{\partial}{\partial x}f(x,y)=y+1$$ $$\frac{\partial}{\partial y}f(x,y)=x+1$$ using Leibniz notation. If you haven't seen this notation, the choice of $x$ or $y$ in the denominator of the $\frac{\partial}{\partial x}$ tells the reader that you are not interested in the other variable. I have never seen the notation $f'(x,y)$ used, and I don't think it would be considered acceptable by most authors. Hope this helps!
P.S. I'm in my final year of high school, so hopefully this isn't hard to understand :)