Implicit Differentiation
Implicit differentiation is a technique used in calculus to compute the derivative of functions even when an explicit formula for the function is unknown. This technique is a consequence of a general theorem, called the implicit function theorem, which enables mathematicians to produce a limitless supply of smooth curves and surfaces.
A variable y is said to be an explicit function of another variable, x, if there is an equation y = f(x) relating y to x. Here f represents a function, in other words a rule that assigns to each value of x one and only one value of y. But the equations of many classical curves, such as the circle x2 + y2 = 1, are not written in the standard y = f(x) format. Instead, x and y are mutually dependent, and so it is not immediately clear whether y can actually be considered to be a function of x.
In fact, in the cited example each value of x corresponds to two values of y, namely y = ((1 - x2) and y = -((1 - x2). Therefore y is not a function of x over the entire curve. However, over a small piece of the circle - for example, a small arc containing the point (3/5, 4/5) - the variable y is indeed uniquely defined as a function of x. It is said to be "implicitly defined" by the equation x2 + y2 = 1.
Not only is y a function of x in this restricted sense, but its derivative can be computed by differentiating the equation for the circle and applying the Chain Rule, to give the new equation 2x + 2y(dy/dx) = 0. Solving this equation for dy/dx gives dy/dx = -x/y. At the point (3/5, 4/5) mentioned above, it follows that dy/dx = -3/4. This method of computing the derivative is much easier than the alternative of solving for y as a function of x and differentiating explicitly.
In general, the implicit function theorem states that an equation g(x, y) = C(where g is a continuously differentiable function and C is any constant) locally defines y as an implicit function of x in a neighborhood of any point where the partial derivative of g with respect to y is nonzero. At any such point, the derivative dy/dx equals -((g/(x)/(( g/(y). The theorem extends as well to equations of three or more variables. It can be shown that an equation g(x, y) = C defines a smooth curve at all points except where both partial derivatives of g are 0, and similarly that an equation g(x, y, z) = C defines a smooth surface at all points except where all three partial derivatives are 0. Such points are called singular points of g.
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