Derivative of a Function

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So far we have discussed the derivative of a function f(x) at a point 'a' which is in the domain of f. Suppose we want to find the derivative of the same function at a different point 'b', then we have to compute the derivative by repeating the same process. To avoid this repetitive process, we can define the derivative of a function for all the points in the domain of f.So far we have discussed the derivative of a function f(x) at a point 'a' which is in the domain of f. Suppose we want to find the derivative of the same function at a different point 'b', then we have to compute the derivative by repeating the same process. To avoid this repetitive process, we can define the derivative of a function for all the points in the domain of f.

Notion for the Derivative of a Function

The derivative of the function f with respect to a variable x is the function f ' whose value at x is

provided the limit exists.

Note:

(i) Note that f ' is a function.

(ii) The domain of f ' is the set of points in the domain D of f for which the limit exists.

(iii) The domain of f ' may be a subset of the domain of f.

(iv) If f(x) exists, we say that f has a derivative at x or f is differentiable at x.

(v) The main difference between f '(a) and f '(x) is that f '(a) is a

number where as f '(x) is a function.

(vi) More often, h is replaced by Dx, where Dx denotes a small change in x. The corresponding change in the function y = f(x) is denoted by Dy.

Example:

If f(x) = x² - 9x + 20, then find f '(x) and hence find f '(100).

Suggested answer:

The function

where Dx is a small change in x.

= 2x - 9

f '(100) = 2 (100) - 9 = 191