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Chain rule is a differentiation technique which can be used in either implicit or explicit differentiation, depending upon the problem. On the other hand, implicit differentiation is a differentiation technique, which is used when all x's and y's are on the same side. Example: x squared + y squared = 4xy, in this case, you use implicit differentiation to actually differentiate the equation, and you use the chain rule to differentiate 4xy.
implicit function is a function in which the dependent variable has not been given "explicitly" in terms of the independent variable. To give a function f explicitlyis to provide a prescription for determining the outputvalue of the function y in terms of the input value x: : :: y = f(x). By contrast, the function is implicit if the value of y is obtained from x by solving an equation of the form: : :: R(x,y) = 0. That is, if it is defined as the level set of a function in two variables: one variable or the other may determine the other, but one is not given an explicit formula for one in terms of the other. Formally, a function f:X→Y is said to be an implicit function if it satisfies the equation : :: R(x,f(x)) = 0 for all x∈X, where R is a function on the Cartesian product X × Y. Every function R defines an isocontour
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