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| Derivative of a Function (in general) |
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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. |
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| The derivative of the function f with respect to a variable x is the function f ' whose value at x is |
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| provided the limit exists. |
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| Note: |
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| (i) Note that f ' is a function. |
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| (ii) The domain of f ' is the set of points in the domain D of f for which the limit exists. |
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| (iii) The domain of f ' may be a subset of the domain of f. |
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(iv) If f (x) exists, we say that f has a derivative at x or f is differentiable at x. |
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| (v) The main difference between f '(a) and f '(x) is that f '(a) is a |
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| number where as f '(x) is a function. |
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| (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. |
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| Example: |
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| If f(x) = x2 - 9x + 20, then find f '(x) and hence find f '(100). |
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| Suggested answer: |
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| The function |
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| where Dx is a small change in x. |
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| = 2x - 9 |
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| f '(100) = 2 (100) - 9 = 191 |
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