Implicit differentiation

In multivariable calculus, we can differentiate implicitly-defined functions using the chain rule. For a function of two independent variables, assuming $f$ is differentiable on its domain, and $f_y\ne 0$, we differentiate assuming $x$ is the independent variable wrt $x$.

$$\frac{\partial f}{\partial x}\frac{dx}{dx}+\frac{\partial f}{\partial y}\frac{dy}{dx}=0$$

A useful simplification is:

$$\frac{dy}{dx}=-\frac{f_x}{f_y}$$

But this is fairly easy to derive, even for functions of more than 2 variables. For some function $F(x,y,z(x,y))=0$, we can differentiate implicitly wrt some variable using the chain rule and isolate, say for $\frac{\partial z}{\partial x}$.

See also

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