00:01
In this question, we're asked to find the maximal directional derivative magnitude and direction for the function f of x, y equals x cubed plus 2xy minus the cosine of pi y at the point 3, 0.
00:13
So first, i'm going to need my partial derivative with respect to x because my gradient of f is comprised of two components, the partial of f with respect to x and the partial of f with respect to y.
00:27
My partial with respect to x is 3x squared plus 2y, while my partial with respect to y will be 2x and then it's going to be plus sine of pi y times pi.
00:48
Now i'm going to evaluate that at the point that i've been given, 3, 0.
00:53
If i do in my first component, i have 3 times 3 squared, 27 plus 0 is 27.
01:01
In my second component, i'm getting 6 plus the sine of 0, that's just 6.
01:09
Now if i want the direction, if i want the direction for the maximal directional derivative, all i do is take that gradient vector, 27, 6, and i convert it into a unit vector.
01:37
Some books will allow you to leave it as 27, 6, but most require you to turn it into a unit vector.
01:44
So i'm going to divide by the square root 27 squared plus 6 squared.
01:50
I'm dividing by the magnitude of that vector...