A derivative is the instantaneous rate of a change of a variable with respect to another variable. For example, Velocity is the rate of change – or derivative – of an object with respect to time. When we say a plane is flying 500 miles per hour from east to west, we are describing the rate of change of its position as time changes.
On a graph, the derivative – or instantaneous rate of change – of a function, looks like a tangent line to the curve of the function at the point of measurement. This makes perfect sense, since the tangent line is pinpointing the exact slope of the function at that moment. But as the function moves along the x-axis, the instantaneous rate of change – or slope – of the function changes. This is what creates the curve of a function, which is nothing more than an infinite number of tangent lines at each point of the curve.
In a Multivariable Function, a Partial Derivative is the function’s instantaneous rate of change – or derivative – with respect to only one of those variables, while the others are held constant.