What is the difference between normal and directional derivatives?

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On pages 22 and 23 of H. K. Dass's Mathematical Physics, I read that both normal and directional derivatives are solved by finding the gradient of a surface and then putting a Cartesian value of a point on that surface in the the answer of gradient. I don't understand what makes them different or why they have different names if they are the same thing. For example, eigenvalues of a matrix have other names but it is specified that they are the different names of the same value.