1.For any function y+f(x), express dy/dx using limit notation. Give a clear explanation for your given answer.
you probably meant to type y=f(x)
By definition
dy/dx = Limit (f(x+∆x) - f(x))/∆x as ∆x-->0
The explanation of this equation is the whole basis and introduction to Calculus.
It can be found in any introductory Calculus text.
To express dy/dx using limit notation for any function y=f(x), we start by considering a small change in x, denoted by ∆x. We want to find how y changes with respect to this small change in x, which is given by the derivative dy/dx.
The definition of the derivative involves taking the limit as ∆x approaches 0. We calculate the difference in the function value, f(x+∆x) - f(x), corresponding to the change in x. Dividing this difference by ∆x gives us the average rate of change.
Taking the limit as ∆x approaches 0 means we are considering an infinitely small change in x. As ∆x becomes infinitesimally small, the average rate of change approaches the instantaneous rate of change, which is the derivative.
Therefore, the expression dy/dx in limit notation is:
dy/dx = Lim(∆x-->0) (f(x+∆x) - f(x))/∆x
The limit notation represents the notion of approaching an infinitesimal change in x, which is the fundamental idea behind calculus. It allows us to calculate rates of change and determine the behavior of functions at specific points.