I can’t figure them out. Pls help
In: Mathematics
a limit is the point that y approaches when x approaches a specified value
or in more formal notation:
a limit is the point that f(x) approaches when x approaches a specified value
the derivative is itself a limit. you should look up the definition for it right now. to understand how the derivative works, look up secant lines then look up tangent lines. i promise you, this is the easiest way to understand derivatives (and the right way)
if you draw a secant line that crosses through two points on a curve, you have a rough estimate of the slope of the curve. since the function we are concerned with is a curve, the slope is different at each point on the curve and we can do better than a rough estimate. that’s where derivatives come in
imagine two points on a curve with one fixed and the other free to move. if you move the second point, you get a new secant line each time. if you move the point right on top of the fixed point, you get the line tangent to the curve at that point. the tangent line shares the same slope as the curve at that point
the process of moving the point is the same as saying the distance between the two points’ x coordinates (h) approaches zero. that’s what the lim h->0 part is saying. the rest of it is just the formula for the slope of the secant line. but the limit of this formula is the slope of the line tangent to curve at that value of x
once you get that understanding, you can then think of the derivative of a function as a new function all its own. if you take the derivative of f(x), you get a new function f'(x). this new function is capable of returning the slope of f(x) for any given value of x
you can use the limit formula to derive f'(x) for any given f(x), provided it exists. i suggest you give it a shot by picking a function (try f(x) = x^2 ) and plugging in the right information into the derivative formula. then solve the limit. you will end up with the derivative of f(x) aka f'(x). the derivative of x^2 is 2x
there are faster ways to find derivatives of functions but the definition is very important to learn and understand
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