DERIVATIVES OF INVERSE FUNCTIONS 5-E Inverse Functions (1) Two functions are inverses if f g ( x) g f ( x) x (2) The graph of f contains the point (a,b) if and only if the graph of the inverse contains (b,a) (3) Reflect over the line y=x Inverse Functions At right are the graphs of a function f(x) and its inverse f-1(x). Do you see a relationship between the slope of the graph of f at (a,b) and the slope of the graph of the inverse at (b,a)? • • To Find Inverses • Solve the equation for x • Interchange x and y • Replace y with f 1 x 1) let f (x) x a) sketch the graph 1) let f (x) x b) find the inverse c) sketch the inverse 1) let f (x) x d) differentiate both f(x) and f-1(x). 1) let f (x) x e) find the slope of the graph of f(x) at (4, 2) and the slope of the inverse at (2, 4) f) What conclusion can you make about these slopes? conclusion Since slope m = dy/dx, it should make sense that switching x and y (for inverse functions) should produce reciprocal slopes for inverse functions 1 2) Find the derivative of the inverse of y x 1 2 a) Find inverse then differentiate Derivative of an inverse 1 f ( x) , If (a, b) is a point on f, then (b, a) is a point on 1 then ( f )' (b) 1 df 1 f (b) dx Or if g is the inverse of f. If f is differentiable then 1 g ( x) f ( g ( x)) Take the reciprocal derivative Proof [ f ( f 1 )' ( x)] x d d ( x) 1 [ f ( f )( x)] dx dx f ' ( f 1 )( x) ( f 1 )' ( x) 1 1 f ' ( f )( x) f ' ( f 1 )( x) ( f 1 )' ( x) 1 1 f ' ( f )( x) dy 1 dx dx dy 3) Find the derivative of the inverse of g ( x) x3 1 at x 9 4) Find the derivative of the inverse of f ( x) x 4 at x 12 2 5) Find the derivative of the inverse of y 2 x 12 at x 14 3 HOME WORK Derivatives of Inverse functions Worksheet