implicit differentiation (original) (raw)

Example: The first step is to identify the implicit function. Suppose we have simplified an equation to the form x2+y2+x⁢y=0 (Since this is a two dimensional equation, all one has to check is that the graph of y may be an implicit function of x in local neighborhoods.) Then, to differentiate implicitly, we differentiate both sides of the equation with respect to x, using the chain rule whenever we encounter y, as y is treated as a function of x. We will get

Next, we simply solve for our implicit derivative d⁢yd⁢x=-2⁢x+y2⁢y+x. Note that the derivative depends on both the variable and the implicit function y. Most of your derivatives will be functions of one or all the variables, including the implicit function itself.