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2 min readβ’december 31, 2020

Sumi Vora

Imagine you have a general function f(x), and you want to find the rate of change of that function over a certain interval [a, b]. Thatβs pretty basic: you would just find the slope of the line that connects those two points.Β π

Another way to think about this is to let a = x, representing any x-value on the domain of the function. As follows, b = x+h, where h is any constant added to the value of x. (For example, if a = 2 and b = 7, then x=2 and x+h = 2+5 = 7)Β

Our formula now becomes:

slope = f(3)-f(1)/(3-1) = (24-0)/2 = 12

Now, what if you want to find the rate of change at **a single point**? Well, of course that is impossibleβ¦ a function canβt be changing at a single point. But, as the points get infinitely closer and closer together, we can see that the slope of our line gets closer and closer to this value.Β

In other words, as h (the difference between the points) approaches 0, the slope of our line approaches the rate of change at a singular point of the functionΒ

A **derivative** represents the __instantaneous rate of change__ of a curve at a single point, and is also represented by the __slope of the tangent line__ to the curve at that point.Β β¨

Thus, the **rate of change** of a function would be represented by:

f'(x) = lim h->0 [f(x+h) - f(x)]/h |

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