Differentiation, Velocity as Limit, Geometrical Interpretation of dy/dx

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Velocity as Limit

Let a particle initially at rest at moves along a Strainght line ., The distance

Strainght line

Strainght Line

Strainght line

Covered by it in reaching is a function of time , We may write distance

In the same way in reaching a point close to covering

i.e., is a fraction of time so that

The average velocity of the particle in the interval is given by

[From (1) and (2)]

(Average rate at which distance is travelled in the interval ).

Now we make smaller to obtain average velocity in smaller interval near . The limit of average velocity as is the instantaneous velocity of the particle at time (at the point ).

Velocity at time

It is denoted by .

Thus, if gives the distance of a moving particle at time , then the derivative of at represents the instantaneous speed of the particle at the point i.e. at time.

This is also referred to as the physical interpretation of a derivative of a function at a point.


The distance meters travelled in time seconds by a car is given by the relation . Find the velocity of car at time seconds.



Velocity of car at any time

Velocity of the Car At

Cal Interpretation of Dy/Dx

Let be a continuous function of , draw its graph and denote it by .

y=f(x) be a Continuous Function

Y=F(X) be a Continuous Function

y=f(x) be a Continuous Function

  • Let be any point on the graph of or curve represented by . Let be another point on the same curve in the neighbourhood of point .

  • Draw and perpendiculars to x-axis and parallel to x-axis such that meets at . Join and produce the secant line to any point . Secant line makes angle say with the positive direction of x-axis. Draw tangent to the curve at the point , making angle with the x-axis.

Now, in

Now, let the point move along the curve towards so that approaches nearer and nearer the point .

Thus, when and consequently, the secant tends to coincide with the tangent .

From (1).

In the limiting case,


  • Thus the derivative of the function at any point on the curve represents the slope or gradient of the tangent at the point .

  • This is called the geometrical interpretation of .

  • It should be noted that has different values at different points of the curve.

  • Therefore, in order to find the gradient of the curve at a particular point, find from the equation of the curve and substitute the coordinates of the point in .

Corollary 1:

  • If tangent to the curve at is parallel to x-axis, then or, i.e., or i.e.,.

  • That is tangent to the curve represented by at is parallel to x-axis.

Corollary 2:

  • If tangent to the curve at is perpendicular to x-axis, or .

  • That is, the tangent to the curve represented by at is parallel to y-axis.

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