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Idea of Derivatives

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Idea of Derivatives - Lesson Summary

The idea of derivatives is connected to the rate of change of a function.

The distance versus time graph of a free falling body is as shown.

The graph represents the function, f(x) = ½gt2 .

Here, 'g' is a real number.

The concept of the derivative comes when we wish to find the velocity at a particular instant.

Finding the velocity of the body after 'a' seconds:

The velocity of a body at any given time is given by the distance covered divided by time.

Velocity = Distance/Time = s/t .

The velocity at 'a' is given by the formula (sa - 0)/(ta - 0) = sa/ta.

If we move from the origin to an instant of time, say t1, the expression for velocity is (sa - s1)/(ta - t1) .

And it is continued. The velocities in each case are

Velocity = (sa - s2)/(ta - t2)

Velocity = (sa - s3)/(ta - t3)

ta - t1 < ta - t2 < ta - t3 < .... < ta - tn

Now we come close to the instant of time, 'ta,' at an instant, say, 'tn'.

tn ≅ ta

If we continue the process to an extent where we bring the interval very close, then we arrive at the derivative.

Velocity = (sa - sn)/(ta - tn)

tn - ta ≅ 0

In this method, the velocity of the body at the instant, a seconds, can be found accurately.


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