Inverse function rule
If y be a function of x, such as y = f(x), we may, by solution of the equation, determine x in terms of y, or producing another equation of the form x = g(y). For example, when y = x2, x = y1/2.
The equations y = f(x) and x = g(y) are connected, being in fact the same relation in different forms; and if the value of y from the first be substituted in the second, the second becomes x = g(fx) or, as it is more commonly written, gfx. That is, the effect of the operation denoted by g is destroyed by the effect of that denoted by f, as in the instances (x2)1/2, elnx, asin(sinx)), each of which is equal to x.
By differentiating the first equation y = f(x), we obtain dy/dx = f'(x), and from the second, dx/dy = g'(y). But whatever values of x and y satisfy the first equation, satisfy the second also; hence, if when x becomes x + dx in the first, y becomes y + dy; the same y + dy substituted for y in the second, will give the same x + dx. Hence, dx/dy as deduced from the second, and dy/dx as deduced from the first, are reciprocals for every value of dx. The limit of one is therefore the reciprocal of the limit of the other.
But dx/dy or g'(y), deduced from x = g(y), is expressed in terms of y, while dy/dx or f'(x), deduced from y = f(x), is expressed in terms of x. Therefore, g'(y) and f'(x) are reciprocals for all such values of x and y as satisfy either of the two first equations.
For example let y = ex, from which x = ln(y). From the first dy/dx = ex; from the second dx/dy = 1/y; and it is evident that ex and 1/y are reciprocals whenever y = ex.