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| Product Rule for Differentiation |
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| Let y = u. v, where both u and v are differentiable functions of x. |
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| or (u.v)' = u.v' + v .u' |
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| It can be remembered as: |
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| 'Derivative of the product of two functions = |
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| first function x derivative of second function + second function x derivative of first function' |
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| Remark: |
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| If we divide by uv in the formula ((u.v)' = u.v' + v .u'), we have |
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| It can be generalised for derivative of the product of more than two functions as given below: |
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| Example 1: |
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| Differentiate xex from the first principles (that is using definition of derivative). |
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| Suggested answer: |
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| Let y = f(x) = xex |
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| = xex (1) + ex(1) |
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| Example 2: |
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| Differentiate the following function using product rule. |
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| y = (3 sec x - 4 cosec x) (2 sin x + 5 cos x) |
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| Suggested answer: |
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| = (3 sec x - 4 cosec x) (2 cos x - 5 sin x) |
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| + (2 sin x + 5 cos x) (3 sec x.tan x + 4 cosecx cot x) |
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| = 6 sec x. cos x - 15 sec x. sin x - 8 cosec x. cos x |
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| + 20 cosec x. sin x + 6 sin x sec x. tan x + 8 sin x. cosec x . cot x + 15 cos x. sec x. tan x + 20 cos x. cosec x. cot x |
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| = 6 - 15 tan x - 8 cot x + 20 + 6 tan2x + 8 cot x + 15 tan x |
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| + 20 cot2x |
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| = (6 + 6 tan2x) + (20 + 20 cot2x) |
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| = 6(1+tan2x) + 20 (1+cot2x) |
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| = 6 sec2x + 20 cosec2x. |
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| Example 3: |
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| Differentiate y = (x+2) (x+3) by using the formula |
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| Suggested answer: |
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| Let u = x + 2, v = x + 3 |
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| = 2x + 5 |
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