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Derivatives of Implicit Functions

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Until now we have been differentiating various functions given in the form y = f(x). But it is not necessary that functions are always expressed in this form. For example, consider one of the following relationships between x and y:
x – y – π = 0 
x + sin xy – y = 0 
In the first case, we can solve for y and rewrite the relationship as y = x – π. In the second case, it does not seem that there is an easy way to solve for y. Nevertheless, there is no doubt about the dependence of y on x in either of the cases. When a relationship between x and y is expressed in a way that it is easy to solve for y and write y = f(x), we say that y is given as an explicit function of x. In the latter case it  is implicit that y is a function of x and we say that the relationship of the second type, above, gives function implicitly. In this subsection, we learn to differentiate implicit functions.

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