Directional Derivatives

Calculus 3 · Axiom Academy

LESSON Directional Derivatives The rate of change of a function in any direction — and why the gradient is the steepest way up. Imagine standing on a hillside described by f(x,y) . You already know two slopes: the rate heading east , , and the rate heading north , . But what is the slope if you set off in some other direction — northeast, or at any arbitrary angle? the rate of change of f at (a,b) as we step a distance h along the direction The requirement is not a technicality. If is a unit vector, then t measures true distance travelled, so g'(0) is a genuine rate of change per unit distance . Feed in a longer vector and you would inflate the answer by its length — a classic mistake. Always normalize first. 3. The Formula: Gradient Dotted with Direction We never have to compute that limit directly. Parametrize the path as x(t) = a + u_1 t , y(t) = b + u_2 t and apply the chain rule to : Geometrically the dot product is a projection : is the length of the shadow the gradient casts onto the line of . Because , that projection is exactly , where is the angle between and . 4. Computing a Directional Derivative Normalize the direction so that is a unit vector. Example: f(x,y) = x^2 + y^2 at (1,1) in the direction Notice step 2: has length 5 , so we divide by 5 before taking the dot product. Skipping that step would have given 14 instead of — five times too large. 5. Steepest Ascent: Turning the Direction

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