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Differentiation of Implicit Functions

If the variables x and y are connected by a relation of the form f(xy) = 0 and it is not possible or convenient to express y as a function x in the form y = φ(x), then y is said to be an implicit function of x. To find dy/dx in such a case, we differentiate both sides of the given relation with respect to x, keeping in mind that the derivative of φ(y) w.r.t. x is 85084.png.
For example,
85078.png = cos y  85072.png = 2y 85066.png
It should be noted that 85060.png = cos y but 85054.png = cos y  85048.png.
Similarly, we have 85042.png, whereas 85035.png.
A direct formula for implicit functions
Let f(xy) = 0. Take all terms to the left side and put left side equal to f(xy). Then

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