Finding Directional Derivative at P(2,1) in Direction <3,4>

Calculus 3 · Axiom Academy

EXAMPLE Directional Derivative of f(x,y)=x^2y+y^3 at P(2,1) Compute using the gradient and a normalized direction vector. Find the directional derivative of f(x,y) = x^2y + y^3 at the point P(2,1) in the direction of the vector . Nicely done. You computed a directional derivative end to end — here is the method to carry into any problem. Gradient first: Build from the partial derivatives with respect to each variable. Evaluate at the point: Substitute the coordinates of P into to get the gradient at that location, . Normalize the direction: The direction must be a unit vector. Divide by its magnitude , giving . Skipping this is the most common mistake. Dot product: , where is the unit direction. Interpretation: , so f is increasing at P(2,1) in the direction of at a rate of 8 units. This systematic approach works for any multivariable function. The directional derivative extends the idea of slope, measuring how f changes in any specified direction.

This is the written version of the interactive lesson above. See the full Calculus 3 course.