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Inverse Function Rule

The following rule shows that the derivatives of inverse functions are always reciprocals of each other.

INVERSE FUNCTION RULE

Suppose f and g are inverse functions, so that the two equations

y=f(x) and x = g(y)

have the same graphs. If both derivatives f'(x) and g'(y) exist and are nonzero, then

02_differentiation-219.gif

that is,

02_differentiation-220.gif

PROOF

Let Δx be a nonzero infinitesimal and let Δy be the corresponding change in y. Then Δy is also infinitesimal because f'(x) exists and is nonzero because f(x) has an inverse function. By the rules for standard parts,

02_differentiation-221.gif

Therefore

02_differentiation-222.gif

The formula

02_differentiation-224.gif

in the Inverse Function Rule is not as trivial as it looks. A more complete statement is

02_differentiation-225.gif computed with x the independent variable

02_differentiation-226.gif computed with y the independent variable.

Sometimes it is easier to compute dx/dy than dy/dx, and in such cases the Inverse Function Rule is a useful method.
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Last Update: 2010-11-25