2.3: Implicit Differentiation
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Equation 2.2.5 can be used to solve the problem of differentiation of an implicit function. Consider, for example, the unlikely equation
ln(xy)=x2y3
Calculate the derivative dy/dx. It would be easy if only one could write this in the form y = something; but it is difficult (impossible as far as I know) to write y explicitly as a function of x. Equation ??? implicitly relates y to x. How are we going to calculate dy/dx?
The curve f(x,y)=0 might be considered as being the intersection of the surface z=f(x,y) with the plane z=0. Seen thus, the derivative dy/dx can be thought of as the limit as δx and δy approach zero of the ratio δy/δx within the plane z=0; that is, keeping z constant and hence δz equal to zero. Thus equation 2.2.5 gives us that
dydx=−(∂f∂x)/(∂f∂y).
For example, show that, for Rquation ???,
dydx=y(2x2y3−1)x(1−3x2y3).