My textbook says it’s not a fraction but defined in terms of the limit of a fraction, what does this actually mean? I’m generally having trouble understanding the notation, particularly in the context of implicit differentiation.
Thanks.
In: 0
it is the derivate of y in regards to x.
if y =x^2 the dy/dx = 2x
The second derivative is d^2 y /dx^2 =2
You could derivate in regards to another variable dy/dt = 0 because y do not depend on t.
The notation is common when you have a function that depends on multiple variables as was created by Leibniz in 1675
It’s read as “the change in y with respect to the change in x”. Basically, it’s asking you how changes in x affect y.
If I asked you how volume changes with respect to pressure, you could graph that out. If I asked you how volume changes with respect to color, the graph would look very different – it doesn’t. It’s important to specify the respected property so you know which graph to draw.
It is the instantaneous rate of change of a function with respect to a variable. It is the change in y with respect to x .
It stands for delta of y divided by delta of x (written as) ∆y/∆x Or in other words. A change in Y divided by the change in X.
Sounds like you’re doing differentials. There’s probably a better place to ask this question if you need help with Calculus. I would try to explain but I struggled with this in school and i don’t want to explain it wrong and leave you worse off. Maybe try Kahn Academy? If that’s still a thing.
Are you familiar with calculus? This is calculus terminology and not division in the sense of a fraction.
For ELI5 terms – we start with something called a “function”, a function is a sort of equation where any possible X value has only one possible Y value. So, if you imagine a graph in your head, a line is a function, a U shaped curve is a function, but a C shaped curve isn’t.
This is calculus, but you can think of the dy or dx symbol as “change in”. So dy/dx is saying “for a given change in X values, what’s the change in y values?”, which we generally call “the slope” of a line.
In the case of a straight line, the slope is constant, so if you use the language Y= mX + C, dY/dX = m and the C gets dropped. So if you have Y = 2x + 5, dY/dX of this function is just 2.
In the case of curves you drop the exponent and multiply to the slope and leave X.
So y=3X^(2) becomes dy/dx = (3*2)x=6x.
Given a funtion y(x), then d(y(x))(h) (or dy for short) is the differential of y(x) which is defined as y'(x)*h where y'(x) is the derivative of y(x) with respect to x and h is a new variable.
The same goes for dx where x = i(x) is the identity function with respect to x. So given x’ = 1 we have dx = h. So dy(x)/dx is just another way to write y'(x).
It generally means “how steep is the slope at this point?”. The notation dy/dx can be summarized as “how much distance (d) do you go up (y) here for every bit of distance (d) you go sideways (x)?”.
So if you have a gentle slope of a lush, rolling hill (or, say, a function y=0.1x+2), then you go up only a little bit when travelling sideways, in this example for every 1 unit of length sideways you only go up by 0.1 of those same units in the upwards direction, so dy/dx is 0.1/1 which is 1/10.
If, on the other hand, you have a sheer cliff of a giant mountain (or, say, y=9x-1), then for every bit you go sideways you go a lot further up, in this example 9 units up per unit sideways, so dy/dx is 9/1 which is 9.
If it’s negative, that means you actually go down, not up.
Since you can easily calculate this for every known function f(x), this becomes a handy tool to find out more about that function. For example, if you want to know the maximum points (peaks) of said function, you simply have to find a point where the slope (dy/dx) first goes up (is positive), then goes down (is negative), i.e. it reaches a peak. That means you just find all the places where dy/dx is zero, and then in a second step you probe whether it went from positive to negative (maximum) or vice versa (minimum).
Imagine you are driving on a high way. Let’s say y is the location of your car and x is the time that you have been driving. You can brake, make turns, or change lines. All these are changes made in your location y as a function of the time x. A fraction of y over x would be your speed. dy/dx doesn’t only describe your speed. It breaks your journey into an infinite amount of time points, and describes how exactly your car would go from point A to point B at any single point of time.
Everyone else already gave the definition in terms of derivative and also slope.
Personally I like this definition from a nice 1914 book of Calculus:
http://djm.cc/library/Calculus_Made_Easy_Thompson.pdf
Page 15 – dy/dx – a little bit of y over a little bit of x
Imagine you have put two points on the line of a graph. You might want to calculate the average slope between these two points, that is how steep a straight line drawn between these two points is.
The way you do this is you take how far the two points are away from each other in the y-axis and divide that by how far the two points are from each other in the x-axis. This is commonly written as Δy/Δx.
Now imagine you start to move one of the points closer to the other one. As they get closer the value of the slope is going to start approaching whatever the slope is at exactly the first point.The problem is that we cannot get the points right on top of each other because then the difference in x-axis between the two points would be zero and we’d end up trying to divide by zero. Instead we see what happens as we get the points closer and closer to each other and observe what value the slope approaches as we do.
This value is what we call the limit of the slope as the distance between the points approaches zero. It’s what we call the derivative of the function in that point. However unlike Δy/Δx it’s not really a proper fraction.It’s what that fraction approaches as we make x infinitely small. And to mark that it’s not actually a proper fraction but rather the limit of a fraction we denote it as dy/dx.
Latest Answers