Inverse functions and differentiation

The inverse of a function <math>y = f(x)</math> is a function that, in some fashion, "undoes" the effect of <math>f</math> (see inverse function for a formal and detailed definition). The inverse of <math>f</math> is denoted <math>f^{-1}</math>. The statements y=f(x) and x=f^-1(y) are equivalent.

Differentiation in calculus is the process of obtaining a derivative. The derivative of a function gives the slope at any point.

<math>\frac{dy}{dx} </math> denotes the derivative of the function <math>y=f(x)</math> with respect to <math>x</math>.

<math>\frac{dx}{dy} </math> denotes the derivative of the function <math>x=f(y)</math> with respect to <math>y</math>.

The two derivatives are, as the Leibnitz notation[?] suggests, reciprocal, that is

<math>\frac{dx}{dy}\,.\, \frac{dy}{dx} = 1 </math>

This is a direct consequence of the chain rule, since

<math> \frac{dx}{dy}\,.\, \frac{dy}{dx} = \frac{dx}{dx} </math>

and the derivative of <math> x </math> with respect to <math> x </math> is 1.

Examples

• <math>y = x^2</math> (for positive <math>x</math>) has inverse <math>x = \sqrt{y}</math>.

<math> \frac{dy}{dx} = 2x