80 - Partial Derivatives

Partial derivatives are very similar to derivatives, the only difference being that only one variable is taken into consideration, with other variables being interpreted as constants, unlike implicit differentiation.

They are written using the delta symbol or the variable under the subscript, And just like the limit definition of a derivative, for a multivariate function the limit definition of the partial derivative is given:

Second Partial Derivatives

Higher order partial derivatives exist too. And they are written in the same notation as derivatives, except we use delta and in the case of the subscript notation we just add another copy of the variable we are differentiating for, so

Second Partial Derivatives are Symmetric

This means that for any variable, we have

80 - Partial Derivatives

Second Partial Derivatives

Second Partial Derivatives are Symmetric

81 - The Gradient