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What are derivatives?

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Ethan Lee

Recently completed an MSc in Mathematics, with teaching experience and training.

A derivative describes the rate of change of a function with respect to a variable. That’s quite the mouthful and without seeing what happens in terms of a picture, there’s no (good) short explanation for what a derivative is! I will do my best though. Suppose you have a graph of y=f(x). Then mark two distinct (different) points P1=(x1,f(x1)) and P2=(x2,f(x2)) on this graph. The ratio between the change in the y-value and the change in the x-value is k = (f(x2)-f(x1))/(x2-x1). As P1 and P2 are taken to be closer and closer together this value of k is the derivative. The difficult part of a wholly written explanation is the last part, which needs you to understand something called limits, whereas if I could show you on a picture, there is an easier (intuition based) explanation.

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