Relationship to Total Derivative
Recall in the article about multivariable derivatives, we talked about how single-variable derivatives approximate a line, and multivariable derivatives approximate a hyperplane.
Orthogonal to the Curve
Direction of Max Increase
If you're an ant on the surface of a unit sphere, and you want to move to a larger sphere, you have an infinite number of options. You can move along the sphere, which means your radius, or $f(x)$ value, would the stay the same. If you move toward the center, the radius decreases. If you move perpendicularly away from the sphere, the radius increases most efficiently. The gradient, the perpendicular direction, goes towards the next highest level curve.
Now, let's show this mathematically $$|D_vf(p)| = |\nabla f(p) \cdot v|$$ We can use the C-S Inequality now. $$|D_vf(p)| = |\nabla f(p) \cdot v| \leq ||\nabla f(p)||||v|| = ||\nabla f(p)|| $$ This means that the maximum value of the directional derivative is when the vector points in the direction of the gradient, which is orthogonal to the curve.
You have one unit sphere at the origin. How far away do you center a sphere of radius 2 so that it's orthogonal to the first sphere?