Submitted by EarlaSallow t3_z774l2 in explainlikeimfive
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.
You actually cannot calculate this for every known function.
Many functions have points where this will not work (e.g any non-continuous functions like the signum function, or non-smooth functions like absolute value) and some esoteric ones (like the Weierstrass function) can't be differentiated anywhere.
It's a pretty fun thought experiment to make up a value and pretend that this value is d(sign(x))/dx at 0, then try to run through some normal calculus with this new constant.
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.
Or, if OP wants to stay on reddit and ask questions: /r/learnmath
There's also a bunch more learning resources posted there.
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.
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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 .
also called Leibniz's notation, When x increases by Δx, then y increases by Δy if you try to make the dx go towards 0 but not exactly 0 you get the rate of change of the sole variable Y
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).
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.
In calculus, it means "changes in y with respect to x". The 'd' is short for 'delta' (the symbol we use for "change" or "difference"), and the ratio of the dy and dx , is the "derivative" or the slope of a line at a points along the line.
Say you had a a line where y = x^(2). That means when x is -2, -1, 0, 1, and 2 that y is 4, 1, 0, 1, and 4 respectively -- it looks like a U-shaped cup. The derivative of that line, dy/dx is 2x. That means at x = -2, -1, 0, 1, and 2, the slope of the line (change in y with respect to x) is -4, -2, 0, 2, and 4 respectively. The line y = 2x + 1 has a derivative dy/dx = 2 -- meaning that the slope is constant all along the line, which is precisely what you expect for a straight line; moreover, it's pretty intuitive, y changes 2 for each 1 that x changes.
Calculus provides a way of figuring out the slopes of lines and the areas underneath them (and it can work with more variables too).
you know how slope for a line is (y2-y1)/(x2-x1)? this is usually written as Δy/Δx. But this only works for straight lines; the slope of a curve changes and so to find the slope at a given point, we can’t measure it across any sizeable Δx.
So what do we do? Well some mathematicians back in the day decided to use their imaginations. dy/dx just means, what if we imagine that x2 gets infinitely closer to x1 without actually being x1? This is dx. Then if y is dependent on an equation of x, let’s say y=x^2, what would be then the difference between y2=x2^2 and y1=x1^2? That would be dy.
You have two limits, and you divide 1 over the other (dy/dx) and if your plot has a smooth curve then the limits will solve out to something. Now extend it out to not just this x1 but for all the possible x in your original equation and you get dy/dx = 2x. This function gives you the slope function of your original function, or in other words, tells you the slope of any point on your original curve.
edit: mixed up an x and y, also some clarity
The "d" in dy/dx essentially means "difference". dy/dx is saying "the difference in y per difference in x", so when x changes a certain amount y changes, like how you might express "50 kilometers/hour" to mean how much change in kilometers per every 1 hour.
However, with that km/h example, you don't want to just be taking the average over a whole hour. Within that time period, you might be going faster or slower at different times. In calculus, we're more interested in the change in y at an exact moment. To do this, we essentially separate x (equivalent to time here) into infinitely small periods, until it's so small we can ignore the length of time. This is what the textbook means by "limit".
With the kilometers per hour example, you could split the hour into 2 sections of 30 minutes, maybe in the first you moved 30km (then went slower in the 2nd half of the hour). 30km/0.5h = 60km/h. You keep doing this, splitting up the time into smaller and smaller pieces until it's infinitely small. If you know an equation that describes the distance you've traveled at any particular time, you can find the exact speed at any individual moment using methods based on this idea.
Conceptually, implicit differentiation works because if one side of the equation equals the other side at all values of x, that means that the other side of the equation would have to be changing at the same rate no matter what the value of x is as well. This means that if you can find out the rate of change of the left side of the equation (d(left side)/dx), it will equal the rate of change of the right side of the equation (d(right side)/dx).
When you find both of these rates of change, you'll end up having the rate of change of y with respect to x in the equation for one or both sides (whichever have y in it to begin with), because the amount that the whole side of the equation changes with a change in x is dependent on how much y changes with that change in x.
It's basically saying that you're differentiating (finding the rate of change) of Y compared to X. For example, if you're driving a car and you know how far you have traveled over time then you can call distance Y and time X then differentiate Distance over Time to get your speed at any point in time. You can differentiate this again to get your Acceleration. The important part is that this isn't just your average speed, it's the formula for calculating your speed at any time over your journey
The Y or X are just the standard names for the variables - you can call them anything you like. In the distance over time example I used, you can call distance D and time T and velocity V then you get V = dD/dT
You could then call acceleration A and get A = dV/dT
The whole idea of calculus is to look at rates of change which is useful for a wide range of applications
ComradeMicha t1_iy539u4 wrote
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).