Hello again, today we shall understand the definition of Differentiation using Limits. For those who have not studied the previous discussion related to the differentiation, I recommend you to go read this post "How well have you understood Differentiation?".
Suppose, we have a curve f(x) and we are trying to find out its derivative at a point. As we have already seen, derivative of a straight line curve f(x) is nothing but the slope of the line. But what if the curve f(x) is not straight? In that case you take the derivative to be the slope of the tangent. Remember that, though the tangent is a straight line, we do not use the definition-"rate change in f(x) to the change of x". Look at the figure below(fig. 1), and observe, we discard the first formula and use the second one defined using the limits.Why so?
Look at the fig. 2.
Observe that if we consider the first definition (refer fig. 1), then we will be calculating the slope of the secant to the curve and not the tangent. But we require the slope of the tangent. How do we calculate the slope of the tangent without actually drawing a tangent to the curve. This is where the "limits" comes handy.
Look at the Animation below (Anim. 1)
In this animation, you see that we have taken a secant to the curve f(x). Note that that as the point (x+dx) moves closer and closer to the point x, the secant line and the curve becomes almost equal, and some stage when dx is very small, the secant gets restricted to only one point on the curve, i.e., 'x' and at this point, it coincides with the tangent of the curve f(x) at the point 'x'. So, we can vaguely say that a secant gradually tends to become a tangent at 'x'.
This is the reason, we take the slope of the secant, using the definition of the slope of straight line. And then in order to calculate the slope of the tangent, which is what we actually require, we apply the concept of limit and find its value, as dx tends to zero.
Hence, by using the concept of limits, we define Differentiation as,
Suppose, we have a curve f(x) and we are trying to find out its derivative at a point. As we have already seen, derivative of a straight line curve f(x) is nothing but the slope of the line. But what if the curve f(x) is not straight? In that case you take the derivative to be the slope of the tangent. Remember that, though the tangent is a straight line, we do not use the definition-"rate change in f(x) to the change of x". Look at the figure below(fig. 1), and observe, we discard the first formula and use the second one defined using the limits.Why so?
 |
| fig. 1 |
Look at the fig. 2.
Observe that if we consider the first definition (refer fig. 1), then we will be calculating the slope of the secant to the curve and not the tangent. But we require the slope of the tangent. How do we calculate the slope of the tangent without actually drawing a tangent to the curve. This is where the "limits" comes handy.
Look at the Animation below (Anim. 1)
This is the reason, we take the slope of the secant, using the definition of the slope of straight line. And then in order to calculate the slope of the tangent, which is what we actually require, we apply the concept of limit and find its value, as dx tends to zero.
Hence, by using the concept of limits, we define Differentiation as,
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