Redbelly98 said:g = GM/r^2 gives the acceleration due to gravity.
a = dv/dt is the basic definition of acceleration (which can be cause be any other force, not just gravity)
Partial differentiation is used when there are more than one independent variables. But since the study of motion in introductory physics uses just one independent variable -- time -- partial derivatives are not relevant.
Differentiation is a mathematical operation that involves finding the rate of change of a function with respect to a single independent variable. Partial differentiation, on the other hand, involves finding the rate of change of a multivariate function with respect to one of its independent variables while holding the other variables constant.
To perform differentiation, you need to take the derivative of a function with respect to its independent variable using the rules of differentiation. For partial differentiation, you will need to take the partial derivative of a multivariate function with respect to one of its independent variables while treating the other variables as constants. This can be done using the chain rule and the product/quotient rule as needed.
The purpose of differentiation is to find the instantaneous rate of change of a function, which can be used to determine important properties such as maximum and minimum values. Partial differentiation is used to find the rate of change of a multivariate function in a specific direction, which is useful in optimization problems.
An example of differentiation would be finding the derivative of the function f(x) = x^2, which results in the function f'(x) = 2x. An example of partial differentiation would be finding the partial derivative of the function f(x,y) = 3x^2 + xy with respect to x, which results in the function fx(x,y) = 6x + y.
The main similarity between differentiation and partial differentiation is that both involve finding the rate of change of a function. Additionally, both use similar rules and techniques, such as the chain rule and product/quotient rule. They are also both important tools in calculus and can be used to solve a variety of mathematical problems.