Table of Contents
What is the derivative rule for fractions?
The Quotient Rule in Words The Quotient Rule says that the derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.
What is the significance of the derivative of a function?
As we have seen, the derivative of a function at a given point gives us the rate of change or slope of the tangent line to the function at that point. If we differentiate a position function at a given time, we obtain the velocity at that time.
Should dy/dx be a fraction or a fraction?
Some would even argue that it allows a more intuitive approach to the calculus, is historically more accurate. As long as this is handled in some way, there is no problem with considering dy/dx as a fraction, or differential quotient. (Which is typically presented instead as a limit, today.)
Do you have to cross multiply dy/dx for integration?
In subjects like integration by substitution and differential equations she said that you had to cross multiply dy/dx to isolate either dy or dx for integration. I always thought dy/dx was a function like a sine or cosine and I don’t see why you can break up dy/dx by cross multiplying. What do dy and dx represent?
What is the problem with dy/dx notation?
The problem is illustrated by the need for the parenthetical remark “under appropriate conditions” in your third paragraph. Using dy / dx means that students have to struggle with both the notion of derivative and the intricacies of the notation.
What is the differential form of DX and Dy?
You can think of x and y as smooth functions on a one-dimensional manifold of states of some system that you are thinking about, then dx and dy are differential forms. In any open region where dx does not vanish we can say that dy / dx is the unique smooth function such that (dy / dx)dx = dy; in other words, dy / dx is dy divided by dx.