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.
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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
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