Understanding directional derivative and the gradient I'm having trouble understanding the proof of directional derivative and the gradient. Could someone give me a easy-to-read proof of the directional derivative and explain why does the gradient point to the direction of maximum increase?
Thank you very much for any help! =)

Although the previous 2 answers are the "real" answers to your inquiry I feel that perhaps you were asking for an intuitive motivation for why grad f represents the maximum increase. The reason lies in the formal reasoning above... Think about what theta being zero really says... that grad f points in the direction orthogonal to (perpendicular) to the contour lines.. higher regions will have contour lines that get closer together and more dense and if grad f if orthogonal to these lines it must be pointing straight up. 
