Review: Inverse Functions If y = f(x) is a one-to-one function, then there is a function y = f -1 (x) … called the inverse function … such that … That is, the inverse function “un-does” the function and vice-versa. Examples ………..
Finding the Inverse of a Function In many cases, the inverse can be found by … 1.Writing the function as y = f(x) 2.Switch the variables … i.e. x = f(y) 3.Solve for y. Examples … find the inverses of …
Finding the Inverse of a Function In other cases, the inverse cannot be found algebraically and therefore the existence of such an inverse is recognized and given a name. Examples …
Derivatives of Inverse Functions What is the relationship between the derivative of a function and the derivative of its inverse? Begin with two simple examples … Reciprocal? Almost!
Derivatives of Inverse Functions The derivative of the inverse of a function is the reciprocal of the derivative of the original function evaluated at the inverse. Or … since the inverse of the inverse is the original function … This may be the more practical form.
Logarithmic Differentiation Due to the laws of logarithms, it is often easier to find the derivative of the log of a function instead of the derivative of the original function. Steps … find the derivative of y = f(x) 1.Take a logarithm of both sides (use natural logs) 2.Expand ln(f(x)) using the laws of logarithms 3.Differentiate implicitly 4.Solve for dy/dx This will simply require multiplying both sides of the equation by y or f(x).