Is there any difference between a "derivative" and a "directional derivative"?

Question

See What is directional derivative?

It is just the usual derivative i.e. the ONE VARIABLE/ONE DIMENSIONAL Derivative obtained from breaking down a VECTOR DERIVATIVE.

This example will make it clear. Consider a humming bird moving around you. You are interested(assume!) in the change in the position of the bird with respect to time(i.e. its velocity). But being a mathematician(assume!), you realize that NOT 1, but actually 3 - Variables: x, y and z are changing. So you call the Velocity in X-direction as the DIRECTIONAL DERIVATIVE OF THE POSITION WITH RESPECT TO TIME IN THE X DIRECTION…

I hope you understand. If not, ponder for a few days. Even if you don’t understand then, just let me know.

See How different is directional derivative from the usual derivative f'?.

All derivatives are directional derivatives, sort of. The directional derivative indicates the rate of change of the function in a specific direction. The usual derivative $f'$ of a function $f$ of one variable indicates the rate of change of the function in the positive coordinate direction. The only other direction that you can define for a function of one variable is the negative coordinate direction, and that turns out to be just $-f'$.

For a function of two variables, you have a lot more choices. There is an infinite number of directional derivatives, one for each direction in the plane. Fortunately, you don’t have to calculate them all separately, it’s enough to compute the directional derivatives in two coordinate directions, the partial derivatives. It is particularly convenient if these direction are chosen to be orthogonal. Then you can obtain the directional derivative in any direction, given by a vector $d$, by forming the dot product between the partial derivatives and the components of the vector $d$ expressed in the same coordinate system.

What I understand is they are the same.

So, Is there any difference between a "derivative" and a "directional derivative"?

Answer 1

Yes, there is a difference. And a big one. Let us assume that $f$ is a differentiable map from $\mathbb{R}^n$ into $\mathbb R$.

The directional derivative of $f$ at a point $p$ with repect to a vector $v$ is a number. That number tells us how fast $f$ grows near $p$ in the direction given by $v$.

The derivative $f'$ at a point $p$ is a linear map from $\mathbb{R}^n$ into $\mathbb R$. And, given a vector $v$, $f'(p)(v)$ is precisely the directional derivative of $f$ at $p$ with respect to the direction provided by $v$.

Answer 2

From the way in which this paragraph is worded it makes most sense to think of a function in more than two dimensions, e.g. $$z=f(x,y)$$ in this context we know that we can take derivatives in two different dimensions, to give: $$\frac{\partial z}{\partial x},\frac{\partial z}{\partial y}$$ when the term derivative is used, we usually mean the of the function with respect to its single variable $\frac{df(x)}{dx}$ or we state in which dimension we intend to find the derivative "the rate of change of $z$ with respect to $x$". From this it is clear to see that we presume the positive direction in any axis ($+x$ as supposed to $-x$). All this concept of "directional derivative" introduces is that we can define a derivative in any direction compared to the function

Answer 3

If you have a function $f=f(x_1,...,x_n)$ which depends on $n$ variables the partial derivative $\frac{\partial f}{\partial x_i}$ is a directional derivative in the direction of the vector $e_i=(0,...,0,1,0,...,0)$ wich has zeros in each entry but the i-th entry where it is $1$.

The directional derivative in a direction $h\in \Bbb R^n$ is defined like this: $$\frac{\partial f(x)}{\partial h}=\lim\limits_{t\xrightarrow[]{}0}\frac{f(x+ht)-f(x)}{t}$$ and is noted $D_hf(x)$. And this definition agrees with the partial derivative with respect to $x_i$ a.k.a. the directional derivative in the direction of $e_i$