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# Derivatives of a Function

### To find the instantaneous speed, we have to use the process of differentiation.  Let’s say we want to find the instantaneous speed at this point ‘P’, that is at the time instant ‘X not’.  For this, we first find the average speed for some time interval ‘delta X’. At ‘X not’ the distance covered will be this. And at ‘X not plus delta X’ it will be this. So the average speed in the time interval ‘delta X’ will be equal to this. For a function, this ratio is called the average rate of change.  Now let’s say we draw a straight line between these two points. It is called the secant line. We see that the slope of this secant line will be equal to the average speed in the time interval ‘delta X’.  So we see that the average rate of change between two values of ‘X’ is equal to the slope of the secant line between the corresponding points on the curve.  Now what in the next step? How do we find the instantaneous speed from this average speed? For this we find the average speed in shorter time intervals, that is, as ‘delta X’ tends to zero.  We say that when the limit ‘delta X’ tends to zero, the average speed approaches the instantaneous speed. .  This instantaneous speed is the instantaneous rate of change at ‘X not’. Now what happens to this secant line as ‘delta X’ tends to zero?  Let’s say we have to find the average rate of change between points ‘P’ and ‘Q one’, ‘Q two’ and so on. We can see that as ‘delta X’ tends to zero, these points come closer and closer to the point 'P'. So as the limit ‘delta X’ tends to zero, the secant line will pass through only one point – ‘P’ on the curve. That is the secant line approaches the tangent line.  This is how using the process of differentiation we find the instantaneous rate of change of a function or the slope of a tangent line. Now here instead of writing the instantaneous rate of change in ‘Y’ with respect to ‘X’ at ‘X not’, we say that the derivative of the function at ‘X not’ is this.The derivative is usually denoted by the symbol ‘D Y by D X’.  So we saw here that the derivative of the function at a particular value of ‘X’ is its instantaneous rate of change at that point.

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