DifferentiationI have learned differentiation, done many problems on it, but couldn't understand what is is exactly. Please give a detailed explanation.

Asked on by cols104

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lfryerda | High School Teacher | (Level 2) Educator

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Differentiation represents either the slope of a function at a point or the rate of change of a quantity at a specific time.  Leading up to grade 12, when you are calculating slope or the rate of change, it is only with straight lines, which you usually learn how to do in grade 9.  Now that you are in grade 12, you find out that you need to calculate slopes of all kinds of different functions.  

Unfortunately, very few functions are straight lines.  In order to calculate the slope of a curved function, you need to use a limiting procedure.  This involves finding the slope of the secant between two points on the function, and then letting the distance between those points go to zero, which gives the slope of the function at that point.  This is done using the formula lim_{h->0} {f(x+h)-f(x)}/h.

Now this is a complicated way of finding the derivative.

There are also simpler ways using some rules that seem complicated to prove but make finding the derivative fairly straightforward with some algebra.

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embizze | High School Teacher | (Level 2) Educator Emeritus

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Differentiation at a point on a graph gives the slope of the tangent line to the graph at that point. Thus it gives the rate of change of the function at a particular point. This allows you to know how the function is behaving at that point -- increasing or decreasing and how fast.

Differentiating a function yields a new function whose values at every point are the slope of the tangent line of the original function. The derivative of y=x^2 is y'=2x. So at the point x=-2 the function y=x^2 is decreasing at a rate of 4:1(y'(-2)=-4) and at the point x=3 the function is increasing at a rate of 6:1(y'(3)=6).

Overall, differentiating gives you the instantaneous rate of change of a function. This could only be estimated prior to the calculus; you could find the average rate of change over an interval, but not the instantaneous rate of change.

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