|Topics in calculus|
In mathematics, to give a function f implicitly is to give an equation R(x,y) = S(x,y) that at least in part has the same graph as y = f(x). It can be useful to define a function f implicitly when there is no simple formula for f(x) so it is not convenient to give its graph in the form y = f(x). If there is a way to rearrange the implicit equation, making the left hand side be y and the right hand side be a formula in x with no ys, then the function can be explicitly defined.
Not every equation R(x,y) = S(x, y) has a graph that is the graph of a function. It might be necessary to use just part of the graph.
This may be true, as in the case of a graph that is a line; it may be true with some limitations, such as specifying that one cannot give a vertical line as a graph; it may be true with some limitations on the function domain, as when the relation is x = C(y) with C a cubic polynomial with a 'hump' in its graph; or it may be true only after also cutting R down to size, as in the case x = y2. That is, an implicit function can sometimes be defined successfully only by modifying the relation by 'zooming in' to some part of the x-axis, and 'cutting back' unwanted function branches. A resulting formula may qualify as an ordinary explicit function.
In calculus, implicit differentiation can be applied to implicit functions. This is by an application of the chain rule, to calculate derivatives dy/dx without necessarily making y an explicit function of x.
Consider for example
- y + x = −4.
- f(x) = y = −x − 4;
such differentiation would result in a value of −1. Equally, one can use implicit differentiation;
- dy/dx + dx/dx = 0 = dy/dx + 1; dy/dx = −1.
An example of an implicit function, for which implicit differentiation might be easier than attempting to use explicit differentiation, is
- x4 + 2y2 = 8.
In order to differentiate this explicitly, one would have to obtain (via algebra)
and then differentiate this function. This creates two derivatives, one for y > 0 and another for y < 0. Implicit differentiation avoids this.
One might find it substantially easier to implicitly differentiate the implicit function;
- 4x3 + 4y(dy/dx) = 0;
- dy/dx = −4x3 / 4y = −x3 / y.
Sometimes standard explicit differentiation cannot be used, and in order to obtain the derivative, another method such as implicit differentiation must be employed. An example of such a case is the implicit function y3− y = x. It is impossible to express y explicitly as a function of x, which means that dy/dx cannot be found by explicit differentiation. Using the implicit method, dy/dx can be expressed;
- 3y2(dy/dx) − dy/dx = 1; factoring out dy/dx shows that
- dy/dx(3y2−1) = 1; which yields the final answer
Implicit function theorem
It can be shown that if R(x,y) is given by a smooth submanifold M in R2, and (a,b) is a point of this submanifold such that the tangent space there is not vertical, then M in some small enough neighbourhood of (a,b) is given by a parametrization (x,f(x)) where f is a smooth function. In less technical language, implicit functions exist and can be differentiated, unless the tangent to the supposed graph would be vertical. In the standard case where we are given an equation
- F(x,y) = 0
the condition on F can be checked by means of partial derivatives.
For the important generalisation to functions of several variables, see implicit function theorem.
For complex implicit functions see http://www.riemannsurfaces.info