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Concept of Derivatives

Concept of Derivative Assignment / Homework Help
The concept of differentiation in calculus deals with finding the slopes of tangents, velocities and other rates of change.

To begin with, let us consider the motion of a particle. Suppose the path followed by the particle is given by the function s = f(t)

As we increase the time by ∆t, the new position assumed by the particle is s + ∆s.

The average rate of change in position of the particle is given by:

change in position
change in time
= (s + ∆s) - s
(t + ∆t) - t
= ∆s
∆t

The velocity v at the given instant is given by   lim       ∆s if the limit exists.
                                                                          ∆t -> 0   ∆t
Therefore, Instantaneous velocity v(t)=   lim       ∆s
                                                                  ∆t -> 0   ∆t


This new function v(t) is called the derivative of function f(t) with respect to t, thus, the velocity is the derivative of the path with respect to time.

Now let's take the example of slope of a tangent line to the given function.

Limits Assignment / Homework Help Consider a function y = f(x) as shown in the graph. PQ is a secant in the curve.

As Q (x + ∆x, y + ∆y) approaches the point P(x, y) along the curve the secant PQ takes the limiting position PT which is the tangent to the curve at point P.

Slope of PQ = (y + ∆y) - y
(x + ∆x) - x
= ∆y
∆x

As Q -> P, secant PQ -> tangent PT and ∆x -> 0

Therefore, Slope of tangent PT =  lim ( Slope of PQ) =  lim    ∆y.
                                                    P->Q                          ∆x->0  ∆x

In the above example if the limit exists we write   lim       ∆y    as   dy and call it the derivative of y with respect to x.
                                                                              ∆x->0    ∆x           dx
Hence, in general derivative gives us the rate of change of a quantity at a given point.

Formal definition of derivative:

The derivative of a given function y = f(x) is defined as the limit of the ratio of the increment ∆y of the function to the corresponding increment ∆x of the independent variable, when the later tends to zero.

In other words,

dy =   lim    ∆y
dx    ∆x->0  ∆x
=   lim   f(x + ∆x) - f(x)
   ∆x->0         ∆x

It is possible for the above limit, not to exist in which case the derivative does not exist. We say that the function y = f(x) is differentiable if it has a derivative.

The process of finding a derivative is called differentiation.

The above formula is also called the first principle of differentiation.

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