properties of the directional derivative

The gradient is a fancy word for derivative, or the rate of change of a function. It’s a vector (a direction to move) that
• Points in the direction of greatest increase of a function
• Is zero at a local maximum or local minimum (because there is no single direction of increase)
The term gradient (grad) typically refers to the derivative of vector functions, or functions of more than one variable. Yes, you can say a line has a gradient (its slope), but using the term gradient for single-variable functions is unnecessarily confusing. Keep it simple.
“Gradient” can refer to gradual changes of color, but we’ll stick to the math definition if that’s ok with you. You’ll see the meanings are related.
Properties of the Gradient
Now that we know the gradient is the derivative of a multi-variable function, let’s derive some properties.
The regular, plain-old derivative gives us the rate of change of a single variable, usually x. For example, dF/dx tells us how much the function F changes for a change in x. But if a function takes multiple variables, such as x and y, it will have multiple derivatives: the value of the function will change when we “wiggle” x (dF/dx) and when we wiggle y (dF/dy).
We can represent these multiple rates of change in a vector, with one component for each derivative. Thus, a function that takes 3 variables will have a gradient with 3 components:
F(x) has one variable and a single derivative: dF/dx
F(x,y,z) has three variables and three derivatives: (dF/dx, dF/dx, dF/dx)
The gradient of a multi-variable function has a component for each direction.
And just like the regular derivative, the gradient points in the direction of greatest increase (trust me on this, I’ll create a derivation a bit later  ). However, now that we have multiple directions to consider (x, y and z), the direction of greatest increase is no longer simply “forward” or “backward” along the x-axis, like it is with functions of a single variable.
If we have two variables, then our 2-component gradient can specify any direction on a plane. Likewise, with 3 variables, the gradient can specify and direction in 3D space to move to increase our function.