SUNY Geneseo Department of Mathematics

Directional Derivatives

Tuesday, October 18

Math 223 01
Fall 2022
Prof. Doug Baldwin

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Directional Derivatives

Based on “Directional Derivatives” in section 3.6 of the textbook.

Key Ideas

The directional derivative equation basically scales rates of change in the x and y (and more, if there are more than 2 variables) directions by the components of a unit vector that defines the direction in which you want a rate of change:

Derivative of F in direction Theta is D F over D X times cosine Theta plus D F over D Y times sine Theta

Examples

Suppose you’re standing at point (0, 0, 0) on the surface of the “egg-carton” function, z = sin x cos y.

Plot of surface made of alternating bumps and hollows

What’s the instantaneous rate of change in z if you move in the x direction? What about if you move in the y direction? What if you move in a direction angled 45˚ to the x axis?

The rates of change as you move in the x and y directions are the partial derivatives with respect to x and y, respectively. The rate of change at a 45 degree angle is the directional derivative with Θ = 45˚.

Partial derivatives are rates of change in X and Y directions; directional derivative is change in other directions

What is the derivative of f(x,y) = 4 - x² - y² in direction ⟨ 1/2, -√3/2 ⟩?

Use the version of the directional derivative equation in terms of the components of the direction vector. But beware that to do this, that vector needs to be a unit vector — which it is in this case:

With unit direction vector, directional derivative is sum of products of partial derivatives and direction components

What about in direction ⟨ -4, 2 ⟩?

Follow the same idea as above, except that this time the direction vector isn’t unit-length. So scale it to unit length first, then plug its components into the equation for a directional derivative.

Scale non-unit direction vector to length 1 and use in directional derivative

Another View

The basic formula for a directional derivative of f(x,y) in the direction of a unit vector ⟨ u_x, u_y ⟩ is D_uf(x,y) = f_x(x,y) u_x + f_y(x,y) u_y.

Can you think of this expression as a dot product? If so, what are the vectors involved in it?

You can, and the vectors are the direction vector u, and a vector of partial derivatives:

Directional derivative is vector of partial derivatives, the gradient, dotted with unit direction vector

That vector of partial derivatives comes up in a lot of contexts, so much so that it has a name: the “gradient.”

It’s the next thing we’ll talk about.

To get ready, please read “Gradient” and “Gradients and Level Curves” in section 3.6 of the textbook.

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