Directional Derivative

So in partial derivatives, you only consider the derivative when changing one of the variable while keeping others fixed. So in case of $f(x,y)$, we you find $\frac{\partial f}{\partial x}$, you are basically cutting the graph at fixed $y$, and now just changing. Similar of partial wrt $y$, you keep x constant.

But in multivariate functions, you can calculate the derivative in multiple directions and not just two. For example you can ask about the derivative in the direction [1,2] ie if $x$changes 1 unit and $y$changes 2 unit.

Imagine 3D graph, with $z=f(x,y)$, now here imagine vertical planes that cuts through the graph. $\frac{\partial f}{\partial x}$ is when you are cutting the graph with a plane at some $y_0$where plane is parallel to $x-z$ plane. Similary for $\frac{\partial f}{\partial y}$. But there are multiple more vertical planes possible that goes cuts trough the graph.

So the directional derivative allows for us to calculate the derivative in the direction of the plane that cuts the graph.

Directional Derivative as a limit

Directional Derivative is basically the rate of change in some particular direction. Basically, when you partial derivative, you measure the delta in output for delta in the input.

Now, in case when the input is multidimensional, i.e it's a vector. You can measure the delta in output when the delta in the input in some particular direction and not just in $x$or $y$ direction.

Hence mathematically, the direction derivative is:

$$\frac{\partial f}{\partial \vec v} = \nabla_{\vec v}f(\vec a) = \lim_{h\to 0} \frac{f(\vec a+h\vec v)-f(\vec a)}{h}$$

So basically this gives you the derivative along the direction $\vec v$. But not here that $\vec v$ just supposed to represent the direction and hence it's magnitude it's sort of irrelevant, but the derivative will change let's say if you have $2\vec v$.

Directional Derivative as a Slope

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