Derivatives and fractions (relationship?)

In summary, the conversation discusses the relationship between derivatives and fractions in calculus, particularly in regards to Leibniz notation and the chain rule. It is noted that while a derivative is not a fraction, it can often be treated like one, leading to useful applications in solving problems. The use of differentials is also mentioned as a way to better understand and use this relationship.
  • #1
phy666
2
0
learning calculus here. got differential calculus, though it is a little foggy, and most of integral calculus, which is a little foggier. also using very unpolished precalc background, though i did give most of it a once-over. i have many questions which i can't think of, but of the top of my head...

when learning leibniz notation it is pointed out that derivatives are different than fractions. However, some similarities, such as the chain rule, point to an obvious relationship. would someone please explain this relationship? compare/contrast? If I had to guess I would say that a derivative, being a quantification (of a function at a certain point), is something like a number, in the same way that a fraction is, and thus(?) owning an analogous internal composition. Does that mean derivatives are subject to algebraic field properties if arithmetic operators are applied? sorry just typing random words here...
 
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  • #2
Slow down - one thing at a time.
 
  • #3
You speaking of the df(x)/dx ? Correct? In which case, it is an operator. You are stating take the derivative of f(x) with respect to x. After that, not quite sure. No differential operator is more like a ring I believe.
 
  • #4
Yes, it is true that [itex]dy/dx[/itex] is NOT a fraction. It is, however, the limit of a fraction so can often be treated like a fraction. For example, it is true that if y= f(x) and [itex]x= f^{-1}(y)[/itex], then
[tex]\frac{dx}{dy}= \frac{1}{\frac{dy}{dx}}[/tex]

You cannot prove that by just turning over the "fraction" in the denominator but you can go back before the limit in "[itex]\lim_{h\to 0} (f(x+h)- f(x)}{h}[/tex]", inverting that fraction and and taking the limit again.

Similarly, we cannot prove the chain rule:
[tex]\frac{df}{dx}= \frac{df}{du}\frac{du}{dx}[/tex]
by "cancelling" the "du"s but we can go back before the limit, cancel and then take the limit again.

That is one reason for the Leibniz notation and for then defining the "differentials" dx and dy separately- so that we can use the fact that, while a derivative is not a fraction, it can be treated like one.
 

What are derivatives?

Derivatives are mathematical tools used to calculate the rate of change or slope of a function at a specific point. They represent the instantaneous rate of change of a function, or how fast the output of a function is changing with respect to its input.

What are fractions?

Fractions are numerical expressions that represent a part of a whole. They consist of a numerator (top number) and a denominator (bottom number) separated by a horizontal line. The numerator represents the number of equal parts being considered, while the denominator represents the total number of equal parts in the whole.

How are derivatives and fractions related?

Derivatives and fractions are related because the derivative of a function can be thought of as the ratio of the change in the output to the change in the input. This can be expressed as a fraction, where the numerator is the change in the output and the denominator is the change in the input.

What is the derivative of a fraction?

The derivative of a fraction is calculated using the quotient rule, which states that the derivative of a fraction is equal to 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.

How are derivatives and fractions used in real-world applications?

Derivatives and fractions are used in various real-world applications, such as in physics to calculate velocity and acceleration, in economics to determine marginal cost and revenue, and in engineering to design and optimize systems. They are also used in finance to calculate interest rates and in statistics to analyze data and make predictions.

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