How well have you understood Differentiation

In our day to day life we are often concerned about the extent to which a change in one quantity affects the change in another quantity. This is called the "rate of change". For example, while you are walking or riding a bike, you might want to know how much a change in the distance you travel affects the time taken to travel. This is called speed.

Differential Calculus is about describing in a precise fashion the ways in which related quantities change. The primary objects of differential calculus are derivative of a function and their applications. Geometrically, the derivative at a point is the slope of a tangent drawn to the graph of the function at that point.


Consider a function, y = f(x).

If f(x) is the equation of a straight line, then there are two numbers m and b such that, y = mx + b, which is called "Slope-intercept form". Here, m is the slope, and we write it as,

Eqn. 1

Fig 1

But, for a general function, y = f(x), which is not a straight line, this definition cannot be used, for it does not have a slope. Slope of such functions cannot be found using this formula. As you see in Fig 1, the slope of such a function at some point, say 'a', is taken to be equal to the slope of the tangent drawn to the function at 'a'. in this case, the formula to find the derivative, or the slope of the curve, can be written as,

Eqn. 2

Observe here, we have used the concept of limits. As stated earlier, slope of the curve is nothing but the slope of the tangent, and tangent is a straight line. So for a straight line the slope of the line is given by the Eqn. 1 above. But in Eqn. 2, we have used limit, which is indifferent from the Eqn. 1. Why?

Now, my question for you is this, where did this limit come from in the Eqn. 2, though we are finding the slope of the tangent which is a straight line. How can you justify it?