Why is gradient the direction of steepest ascent?
In this question and it's many answer, people argue gradient is direction of steepest ascent by defining a random unit vector and then saying that dot product of this with gradient is maximum when both of them are in the same direction. But how does this prove that gradient is direction of steepest ascent
And, it's not just me, the comments also involve asking the person who answered how the heck his arguement proves it. So, why is it so?
Further in the book, Electricity and magnetism by purcell, in page 64 end to page 65 start, he talks of gradient of a function dependent only distance from origin i.e a radial function f(r) and argues the shortest step which we can make to change f(r) to f(r+dr) is move in the radial direction. Ok, this some what makes sense to me but how do I extend this intuition for regular derivatives and further use this to understand the answer I showed before?