What is difference between partial differentiation and total differentiation?

Answer 1

We use differentiation to find the rate at which a function changes as its input changes. This can give us information about the rate at which a physical process is occurring, or about how a physical quantity changes with position, for example. Partial differentiation gives us an extra facility: it is a way for us to find out about the change in a function that depends on more than one input. In real problems, physical quantities very commonly depend on more than one physical variable and we need to know how the quantity changes as we change any of these variables. For example, the sediment build-up on a river bed may be described by a function representing the thickness of the sediment. This function will depend on one or more spatial coordinates (i.e. whereabouts on the bed we look) and will also depend on time. That means we can ask two quite different questions about the sediment thickness: how rapidly does the sediment thickness change as we move over the bed, at any particular time, or how rapidly does the thickness change in time, at any particular point on the bed. Notice that these questions are about two totally different physical characteristics of the sediment build-up. The main point to remember about those two questions is the following. When we are concerned about how the thickness changes as we change one of the variables, we want to keep the other variable fixed. So if we look at different positions we do it at a particular time and if we're looking at different times we do it at a fixedposition. That idea is at the heart of the process of partial differentiation.