Y = x (x-1) (x-2) (x-3) ... (x-n) The nth derivative is (n+ 1)! x-n(n+ 1)/2

Observe that the highest term of y = x (x-1) (x-2) (x-3) ... (x-n) is x (n+ 1).

Find the n-order derivative and it becomes (n+ 1)! x

The second highest degree term is-(1+2+3+) ...+n) x n.

After taking the n-order derivative, the coefficient becomes -n(n+ 1)/2.

So the n derivative of y is (n+ 1)! x-n(n+ 1)/2

Y = x (x-1) (x-2) (x-3) ... (x-n) The nth derivative is (n+ 1)! x-n(n+ 1)/2

The meaning of derivative:

The derivative of a function at a certain point describes the rate of change of the function near that point. If the independent variables and values of the function are real numbers, then the derivative of the function at a certain point is the tangent slope of the curve represented by the function at that point.

The essence of derivative is the local linear approximation of function through the concept of limit. For example, in kinematics, the derivative of the displacement of an object with respect to time is the instantaneous velocity of the object.