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Physics LibreTexts

2.3: Implicit Differentiation

  • Page ID
    7216
  • Equation 2.2.5 can be used to solve the problem of differentiation of an implicit function. Consider, for example, the unlikely equation

    \[ \ln ( xy) = x^2 y^3\]

    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 2.3.1 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

    \[ \frac{dy}{dx} = - \left( \frac{ \delta f}{ \delta x} \right) / \left( \frac{ \delta f}{ \delta y} \right).\]

    For example, show that, for equation 2.3.1,

    \[ \frac{dy}{dx} = \frac{y(2x^2y^3-1)}{x(1-3x^2y^3)}.\]