Step By Step Calculus » 10.8 - Implicit Differentiation

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Synopsis
Implicit Differentiation is the method we use to find dydx when the function y(x) is defined implicitly. Consider the following three examples:
\displaystyle x^2+y^2=4,\qquad y^2=x, \qquad xy-y^2-5=0.
\displaystyle x^2+y^2=4,\qquad y^2=x, \qquad xy-y^2-5=0.
For each of these equations we can consider y=y(x)y=y(x) as being defined implicitly by the equations, but we cannot express yy as a single explicit function of xx, i.e. y=f(x)y=f(x), and so cannot directly compute \dfrac{dy}{dx}\dfrac{dy}{dx}. However, we can still find \dfrac{dy}{dx}\dfrac{dy}{dx} by Implicit Differentiation.
To find \dfrac{dy}{dx}\dfrac{dy}{dx} through implicit differentiation, one simply needs to differentiate both sides of the equation with respect to xx considering yy as a differentiable function of xx.
One can also use implicit differentiation to find higher order derivatives.
Sometimes implicit differentiation becomes simpler if we first take the log of both sides of the equation and then differentiate the resulting equality.