We have already defined the derivative of a function f(x) at a particular point 'a' and derivative of f(x) in general for the variable x as f (a) and f (x) respectively. The restriction in both the cases is that 'the limit must exist'. If |
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| does not exist, then we say that the function is not differentiable. |
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| If the above limit exists, we say the function f(x) is differentiable. |
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| In order to test the differentiability of a function at a point, the right hand derivative and left hand derivatives are introduced as follows: |
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| Let f be a function of x (y=f(x)). Let a be a point in the domain of f. The RHD of f at a is defined as |
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| where h>0, provided the limit exists. |
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| The LHD of f at a is defined as |
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| where h>0, provided the limit exists. |
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| Note: |
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| In the above definition, substitute a + h = x, then h = x - a as |
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| Rf '(a) can be rewritten as |
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| Similarly, substitute a - h = x. Lf '(a) can be written as |
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| A function f(x) is said to be differentiable at a point 'a' if |
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| (i) both Rf '(a) and Lf '(a) exists and finite. |
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| (ii) Rf '(a) = Lf '(a) |
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| Example: |
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Consider the function y=|x|. This function is differentiable on ( ,0) and (0, ), but not differentiable at x = 0. |
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| For x>0, we have |
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| Since the limit exists, f(x) is differentiable at x>0. Similarly, we can show that f(x) is differentiable at x<0. |
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| We shall find RHD and LHD of f(x) at x = 0. |
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| = -1 |
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| Therefore y = |x| is not differentiable at x = 0. |
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| The following theorem say 'Differentiability implies continuity". |
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| Theorem: |
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| If a function f is differentiable at x = a, then it is continuous at x = a. |
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| f is differentiable at x = a. |
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| Now, we have |
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| f(x) is continuous at x = a. |
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| Note: |
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| Every differential function is continuous, but every continuous function is not differential. |
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| We have already shown that y = |x| is not differentiable at x = 0. It can be easily seen that y = |x| is continuous at x = 0. |
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