What is the function derivative?

What happens when a function, say a velocity v(x,y,z,t) has the total derivative with respect to 't' equals to zero, but the partial derivative with respect to t different of zero?

• What is the difference between both derivatives? What happens if a body in a specific position has dv/dt = 0 , but {\partial v}/{\partial t} !=0 ?

• Answer:

  The difference between a partial derivative and the total derivative is that the partial derivative holds all the other variables constant, in effect treating them as independent of t, while the total derivative allows all the variables in the function to be dependent on t. Maybe an example will help clear this up. Suppose you have a function f(x,y) = x+y . Then the partial derivative of f with respect to x is just 1. But now assume that y is a function of x also, say y=x. Now the partial derivative of f holds y constant, so it is still 1. However, the total derivative is: \frac{df}{dx} = \frac{\partial{f}}{\partial{x}} + \frac{\partial{f}}{\partial{y}}\frac{dy}{dx} = 1 + (1)(1) = 2 Now what's the intuition here? We have y=x. So basically our function is f(x) = 2x. and its derivative with respect to x is 2. But because the partial derivative treats y as a constant, it doesn't capture the additional rate of change in f that occurs because y is a function of x.