What is the difference between partial derivative and derivative?

Answer 1

Say you have a function of a single variable, f(x). Then there is no ambiguity about what you are taking the derivative with respect to (it is always with respect to x).

But what if I have a function of a few variables, f(x,y,z)? Now, I can take the derivative with respect to x, y, or z. These are "partial" derivatives, because we are only interested in how the function varies w.r.t. a single variable, assuming that the other variables are independent and "frozen".

e.g., Question: how does f vary with respect to y? Answer: (partial f/partial y)

Now, what if our function again depends on a few variables, but these variables themselves depend on time: x(t), y(t), z(t) --> f(x(t),y(t),z(t))? Again, we might ask how f varies w.r.t. one of the variables x,y,z, in which case we would use partial derivatives. If we ask how f varies with respect to t, we would do the following:

df/dt = (partial f/partial x)*dx/dt + (partial f/partial y)*dy/dt + (partial f/partial z)*dz/dt

df/dt is known as the "total" derivative, which essentially uses the chain rule to drop the assumption that the other variables are "frozen" while taking the derivative.

This framework is especially useful in physical problems where I might want to consider spatial variations of a function (partial derivatives), as well as the total variation in time (total derivative).

Answer 2

A partial derivative is taken with respect to one variable while holding all other variables constant, whereas a derivative is taken with respect to one variable in general. Partial derivatives are used in multivariable calculus to analyze functions of multiple variables.