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.
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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).
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