I factored and simplified dy/dx of
192x^5 + 96x^3 + 12x all the way down to
x^2 = u = (-1/2) and (-1/6). How does the result illustrates part 1 of the Calculus Fundamental Theorem?
The result you obtained, which is x^2 = u = (-1/2) and (-1/6), does not directly illustrate part 1 of the Calculus Fundamental Theorem. Let me explain how the result relates to the fundamental theorem and clarify its connection.
The Fundamental Theorem of Calculus (FTC) consists of two parts. Part 1 states that if F(x) is any antiderivative of f(x), then the definite integral of f(x) from a to b is equal to F(b) - F(a), where a and b are the lower and upper limits of integration, respectively.
To explain the connection to your result, we need more information on the context or original problem you are working on. The expression you provided seems to be the result of finding the derivative dy/dx of a function or expression. However, the FTC deals with integrals, not derivatives.
In order to demonstrate the Fundamental Theorem of Calculus in action, we typically need to start with an original function and then find its antiderivative. The resulting antiderivative, which we'll call F(x), is a function that, when differentiated, will give us the original function.
Once we have the antiderivative, we can apply part 1 of the FTC to evaluate definite integrals. The limits of integration, a and b, in this case, determine the interval over which we are integrating F'(x) (the derivative of F(x)) back to F(x).
So, if you provide the original function or a clearer explanation of the problem, we can further discuss and demonstrate the application of the Fundamental Theorem of Calculus.