The directional derivative is largest in the direction perpendicular to the gradient.

False

False. $D_u f = \nabla f \cdot u = |\nabla f|\cos\theta$, which is maximal at $\theta = 0$, along the gradient, not across it. Perpendicular to the gradient the derivative is zero, which is exactly why that direction traces the level curve. It is also why gradient descent steps across contours rather than along them.