Loss of differentials. This shows up both in differentiation and in integration. The "loss of differentials" is much like the "loss of invisible parentheses" discussed earlier in this document; it is a type of sloppy writing in intermediate steps which leads to actual errors in the final answer.

When students first begin to learn to differentiate, they are always differentiating with respect to the same variable, and so they see no reason to mention that variable. Thus, in differentiating the function y = f(x) = 7x³+5x, they may correctly write

The difficulty, of course, shows up when we arrive at the Chain Rule. Suddenly, the question is no longer "What is the derivative of y", but rather, "What is the derivative of y with respect to x? with respect to u? How are those two derivatives related?" The student who does not make a habit of distinguishing between dy/dx and dy/du in writing may also have difficulty distinguishing between them conceptually, and thus will have difficulty understanding the Chain Rule.

This also leads to difficulties with the "u-substitutions" rule, which is just the Chain Rule turned into a rule about integrals. For instance:

For the first three problems, the student is attempting to use the formula ò (1/u)du = ln |u|+C (which is a correct formula, but not directly applicable). However, the student has learned it incorrectly as  " ò (1/u) = ln |u|+C." Substitute u = 1+x² or u = x³ or u = cos x into that formula to get the first three erroneous answers in the table above. The expressions ò (1/u)du and ò (1/u)dx have very different meanings, but you're likely to confuse them if you write them both as ò (1/u).

For the last problem in the table above, the student is attempting to use the formula ò u²du = (1/3)u³+C, which is a correct formula, but not relevant to the present problem. The student has probaby memorized that formula in the incorrect form ò u² = (1/3)u³+C. The expressions ò u²du and ò u²dx have very different meanings, but you're likely to confuse them if you write them both as ò u².

Another correct way to write the rule about logarithms is . Since this expresses everything in terms of the variable x, it may make errors less likely. Admittedly, it is a complicated looking formula, but it is preferable to a wrong formula. The first, third, and fourth problems in the preceding table all require more complicated methods; just using logarithms won't solve the problems for you. The problem of integrating x ^?3 actually requires a less complicated method -- i.e., without logarithms.

We should prohibit students from writing an integral sign without a matching differential. Just as any "(" must be matched with a ")", so too any integral sign must be matched with a "dx" or "du" or "dt" or whatever. The expression is unbalanced, and should be prohibited. If we're considering a substitution of u = 1+x², then ò (1/u)du is very different from ò (1/u)dx, and so the expression ò (1/u) is ambiguous and meaningless. If you write ò (1/u) in one of your intermediate steps, you may forget whether it represents ò (1/u)du or ò (1/u)dx, and you may inadvertently switch from one to the other -- thus replacing one mathematical quantity with another to which it is not equal.