I'm not sure how to compute the directional derivative of some multivariable function, but I've come across an exercise which says "this is an example of a directional derivative" before they are actually introduced.
The exercise is as follows;
Determine the slope of the surface $z = x^4 − y^2$ at the point $(1, 1)$ in the general direction that makes an angle $\theta$ with the positive $x$-axis. Find a value of $\theta$ with $0 ≤ \theta ≤ \frac{\pi}{2}$ where this slope is zero. This is an example of a directional derivative.
As there is an angle involved I thought I might transform to polar coordinates and then differentiate with respect to $r$ and $\theta$ but I'm not entirely sure where this would lead. Differentiating $z$ w.r.t. $x$ and $y$ and evaluating at $(1, 1)$ gives I believe $4$ and $-2$ respectively, but I'm (again) not sure how to interpret this result. Can anybody give me a hint/explain how I might go about solving a problem like this without simply giving a definition of the directional derivative?
HINT: If you slice in direction $\theta$, you are looking at points of the form $(1+t\cos\theta,1+t\sin\theta)$. Letting $f(x,y)=x^4-y^2$, can you find the slope of the curve $z=f(1+t\cos\theta,1+t\sin\theta)$ at $t=0$?