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Home Vector Calculus Directional Derivatives and Gradients Theorem 1: Directional Derivatives (Two Variables)  
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Theorem 1: Directional Derivatives (Two Variables)
THEOREM 1 Suppose z = f(x,y) is smooth at (a, b). Then for any unit vector U = cos αi + sin αj, the directional derivative f_{U}(a, b) exists and PROOF Let U = cos αi + sin αj. Write x, y, and z as functions of f, x = α + t cos α, y = a + t sin α, z = f(a + r cos α,b + t sin α). Then by the Chain Rule,


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