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| Derivative of Implicit Functions |
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| Till now, the functions that we have discussed, are explicitly functions of x. We have defined y in terms of x. Suppose we have an equation f(x,y) = 0, which cannot be put in the form of y=f(x) to differentiate in the usual way, we can still differentiate the equation f(x,y) = 0. This function in which y cannot be expressed in terms of x, is called an implicit function. |
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| x3+ y3- 9xy = 0 is an example of an implicit function. |
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| The process of differentiating implicit function is called implicit differentiation. |
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| Rules for Implicit Differentiation |
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| Step 1: |
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| Step 2: |
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| Differentiate the terms containing x, y or both xy with respect to x. |
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While differentiating the terms containing y or power of y, first  |
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| Step 3: |
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| Step 4: |
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| Example: |
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| Differentiate the following implicit equation |
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| ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 |
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| Step 1: |
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| ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 |
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| Step 2: |
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| Differentiating with respect to x, we have |
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| Step 3: |
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| Step 4: |
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