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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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).
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