Finding dndxn(x3cos⁡(x)):

Sure, let's find the derivative of the function f(x) = x^3*cos(x). We will use the product rule for differentiation, which states that the derivative of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.

Step 1: Identify the two functions in the product. Here, the first function u(x) = x^3 and the second function v(x) = cos(x).

Step 2: Find the derivatives of these two functions. The derivative of u(x) = x^3 is u'(x) = 3x^2. The derivative of v(x) = cos(x) is v'(x) = -sin(x).

Step 3: Apply the product rule. The derivative of f(x) = u(x)*v(x) is f'(x) = u'(x)v(x) + u(x)v'(x). Substituting the functions and their derivatives gives f'(x) = 3x^2cos(x) + x^3(-sin(x)).

Step 4: Simplify the expression. The derivative of the function f(x) = x^3cos(x) is f'(x) = 3x^2cos(x) - x^3*sin(x).