MTH 251 – Differential Calculus Chapter 3 – Differentiation Section 3.8 Derivatives of Inverse Functions and Logarithms Copyright © 2010 by Ron Wallace, all rights reserved.

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.

Derivatives of Inverse Functions Proof

Derivatives of Inverse Functions Example … NOTE: Where we really need this is when we cannot algebraically find the inverse of a function (i.e. exp, log, trig, & others).

Derivative of the Natural Log Since the inverse of the natural log is the exponential function …

Derivative of the Natural Log Second form ? Note: x < 0 Using the chain rule.

The Chain Rule and the Derivative of the Natural Log Examples … Memorize this one!

Derivatives of other Logs Change of base formula. ln a is just a constant Yes … you need to memorize this one too!

Review: Laws of Logarithms

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).

Logarithmic Differentiation Example 1 of 3 …

Logarithmic Differentiation Example 2 of 3 … (a more practical application)

Logarithmic Differentiation Example 3 of 3 … (a more practical application)

The Power Rule … One Last Time! What if n is a real number?