[SOLVED] Derivative Vectors/Gradient

Problem:

For each of the following functions, find the derivative vector for those points where it is defined:

a.

c.

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a. I know that . As a result, then

So, if

By taking the partial derivatives, then

....

So, the gradient would be

However, I have a hard time understanding the gradient vector/derivative vector. I'm slightly confused because I thought was called the gradient and not a vector. If I can just understand this, then I can do (c) as well.

Thank you for your time.