I am learning multivariable calculus and when I studied directional derivatives I get the idea of partial derivatives fully. But when it gets to directional derivative, I get the idea conceptually that it is rate of change of function in the direction of unit vector. I also get the definition of directional derivatives as limits.
Duf(x,y) = lim h→0 [f(x+ah,y+bh)-f(x,y)] / h -----> eq1
but when it gets to the definition where it says that to find directional derivative multiply x and y components of unit vector (in the direction in which you want to find rate of change of function) with respective partial derivatives and then add both? Mathematically, in case of function of two variables
Duf(x,y) = ∂f/∂x a + ∂f/∂y b ----> eq2
where a and b are x and y component of unit vector in the direction in which we want to find rate of change of function.
So now I want to understand intuitively about following
- mainly, why we add in eq2?
- what is meaning of that partial derivatives are linear?
- how we can get from eq1 to eq2.
I shall be extremely thankful for bridging this gap which is bothering me. Graphical explanation will be helpful!