A Derivation of Steepest Descent Direction

Consider a continuously differentiable function / = /(x) where x is the variable vector 

consider the problem of minimization of / by changing x by dx of length ds along a direction. For a decrease in / due to the change, the rate d//ds has to be negative. The objective is to find the direction or a unit vector w that results in the most negative rate d//ds or, equivalently, minimizes the rate, which from the last equation is 

This problem is subject to the constraint that w, being a unit vector, must have unit magnitude, i. e., 

Applying the Lagrange Multiplier Rule, the equivalent problem is to find the w that minimizes 

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