Anti-Chain Rule Name:

The most fundamental rule for computing derivatives in calculus is the Chain Rule, from which all other differentiation rules are derived. In words, the Chain Rule states that, to differentiate f(g(x)), differentiate the outside function f and multiply by the derivative of the inside function g, so that d f(g(x)) = f 0(g(x)) · g0(x). dx The Chain Rule makes computation of nearly all derivatives fairly easy.

It is tempting to think that there should be an Anti-Chain Rule that makes antidiffer- entiation as easy as the Chain Rule made differentiation. Such a rule would just reverse the process, so instead of differentiating f and multiplying by g0, we would antidifferentiate f and divide by g0. In symbols,

Z F (g(x)) f(g(x)) dx = + C. g0(x)

Unfortunately, the Anti-Chain Rule does not exist because the formula above is not Z always true. For example, consider (x2 + 1)2 dx. We can compute this integral (correctly) by expanding and applying the power rule: Z Z (x2 + 1)2 dx = x4 + 2x2 + 1 dx x5 2x3 = + + x + C 5 3 However, we could also think about our integrand above as f(g(x)) with f(x) = x2 and g(x) = x2 + 1. Then we have F (x) = x3/3 and g0(x) = 2x, so the Anti-Chain Rule would predict that

Z F (g(x)) (x2 + 1)2 dx = + C g0(x) (x2 + 1)3 = + C 3 · 2x x6 + 3x4 + 3x2 + 1 = + C 6x x5 x3 x 1 = + + + + C 6 2 2 6x Since these answers are different, we conclude that the Anti-Chain Rule has failed us.

However, the Anti-Chain Rule does work sometimes. for example, consider Z (2x + 1)2 dx.

Dr. Prince 1 Fall 2014 Anti-Chain Rule Name:

We can compute this (correctly) by expanding and applying power rule:

Z Z 4x3 (2x + 1)2 dx = 4x2 + 4x + 1 dx = + 2x2 + x + C 3

With f(x) = x2 and g(x) = 2x + 1, we have F (x) = x3/3 and g(x) = 2, so the Anti-Chain Rule would predict that

Z F (g(x)) (2x + 1)2 dx = + C g0(x) (2x + 1)3 = + C 3 · 2 8x3 + 12x2 + 6x + 1 = + C 6 4x3 1  = + 2x2 + x + + C 3 6 4x3 = + 2x2 + x + D 3 where we use a new constant D which absorbed a 1/6. Since the two answers above are equal, the Anti-Chain Rule has worked in this case.

Your assignment is to determine when the Anti-Chain Rule works and when it does not. That is, you are to give a rule that would allow someone to easily tell, before any com- putations are done, whether the Anti-Chain Rule will produce the correct antiderivative of f(g(x)) or not. You must prove mathematically that your rule is correct.

Dr. Prince 2 Fall 2014