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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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
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