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How do I find dy/dx for the following equation: (x^2) + 2y = 4(y^3) + lnx?

This is an example of implicit differentiation with respect to x.

The technique for differentiating such an equation is as follows:

1. Differentiate each term in x with respect to x.

2. For each term in y, differentiate with respect to y, and multiply the result by dy/dx.

3. Rearrange the resulting equation, to make dy/dx the subject of the formula.

Solution:

Let equation (x^2) + 2y = 4(y^3) + lnx be called (*).

Differentiating (*) with respect to x, then rearranging, according to the three rules above, gives:

2x + 2(dy/dx) = [12(y^2)](dy/dx) + (1/x) =>

[2 - 12(y^2)](dy/dx) = (1/x) - 2x =>

2[1 - 6(y^2)](dy/dx) = [1 - 2(x^2)]/x =>

dy/dx = [1 - 2(x^2)]/(2x[1 - 6(y^2)]).

 

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