14. Gradient and Directional Derivatives

Right now, we can find the slope on a surface in the $\large x$- and $\large y$-directions, but there are 360 degrees of directions. How do we get all the other directions?

The Gradient

The gradient is an important multivariable derivative:

$\large \nabla f(x,y) =\left< f_x(x,y),f_y(x,y)\right>$

The $\large \nabla$ symbol denotes a gradient (the symbol is called a "nabla"). As we see, the gradient takes a multivariable function and spits out a vector containing the two partial derivatives.

The Directional Derivative

The directional derivative of $\large f(x,y)$ gives us the slope of $\large f$ in any specified direction. We specify the direction with a unit vector $\large \hat{u}$ in the $\large xy$-plane. The directional derivative of $\large f(x,y)$ in the direction of $\hat{u}$ is:

$\large D_{\hat{u}}f(x,y)=\nabla f(x,y)\cdot \hat{u}$

The gif below shows lines in the direction of the directional derivative at a point on a surface for all 360 degrees of directions:

Since this is a dot product, it is a scalar. Here's an example of how to compute a directional derivative:

Tip: don't forget to convert the direction vector to a unit vector!

Tomorrow's class:

• Practice taking gradients and directional derivatives
• The geometry of the gradient

Section Outcomes:

I can:

• Find a gradient or a directional derivative
• Find the directions of steepest ascent/descent at a point on a surface
• Interpret the directional derivative as a slope in a given direction

Other resources

For more on these derivatives, see https://openstax.org/books/calculus-volume-3/pages/4-6-directional-derivatives-and-the-gradient