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| Derivative of Inverse Trignometric Functions |
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| Before finding the differentiation of inverse trigonometric functions, recall how the inverse trigonometric functions are defined and what the domain and range of each inverse trigonometric function. For ready reference, the domain and range of these functions are tabulated below. |
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| Whenever we say differentiability of these functions we consider them in their respective domains. |
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| Let y = sin-1 x, then sin y = x. |
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| Differentiating both sides with respect to x, we get |
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| Hence, we have |
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| Let y = tan-1x. Then tan y = x. |
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| Differentiating both sides w.r.t. x, we get |
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| Differentiating both sides with respect to x, we get |
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| Hence, we have |
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| Let y = sec-1x. Then sec y = x. |
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| Differentiating w.r.t. x, we have |
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| Similarly, |
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| Example 1: |
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| Suggested answer: |
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| Let |
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| Differentiating with respect to x, we get |
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| Example 2: |
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| Differentiate cos-1(2x+3) from first principles. |
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| Suggested answer: |
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| Let y = cos-1(2x+3) |
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| 2x + 3 = cos y ……….(1) |
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| Let Dx be an increment of x and Dy be the corresponding increment of y. |
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| Therefore 2(x + Dx) + 3 = cos (y + Dy) |
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| 2x + 2Dx + 3 = cos (y + Dy) ……….(2) |
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| Subtracting (1) from (2), we have |
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| 2x + 2Dx + 3 - 2x - 3 = cos (y + Dy) - cosy |
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| Using the formula, |
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