Скачать или смотреть Proof of the Directional Derivative Formula, Real Analysis

Let's prove that if f : R^n to R is a differentiable function, then the directional derivative of f at x=a in the direction of a unit vector u can be computed as the dot product of the gradient of f at and u: ∇f(a) · u. This method simplifies the calculation of directional derivatives, sidestepping the need to employ the limit definition.

This result flows naturally from the differentiability of the function. Differentiability assures us that as our input value approaches the point, the difference between the actual value of the function and its linear approximation (like the tangent line or plane) relative to the distance to the point approaches zero. In other words, the function's value is closely mirrored by its linear approximation at that point.

If our function is differentiable, we can approach the point a along the direction specified by the unit vector u. By substituting the expression for the input value with one that includes a small step in the direction of the unit vector and letting that step go to zero, we find our directional derivative.

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