Common errors

cannot stand alone. means the “rate of change of something with respect to x”. If you omit it, the expression means “the rate of change of nothing with respect to x”. This makes no sense and is neither something measurable nor of any interest to us.

Concavity The only thing about concavity that interests us is weather a function is concave up or down. We don’t measure the concavity. So if your answer to a question includes something like “The concavity is increasing” or “The concavity is positive”, consider revising it immediately. Also

Functions with a multiple personality disorder There can be only one in a particular problem and its solution. If you have, for example,, , and in the same solution, you can know immediately that your solution is incorrect. I have been tolerating this so far, but I do not guarantee that I will do so in the future. There are enough letters in the english alphabet. You can name up to 26 functions just using them. I have even seen renaming of a function given in the problem in order to use its name to name another function. Instead of that just give the new function a new name. All the rules of mathematics (including the product, quotient, and chain rule) apply regardless what name you give to a function.

Standalone expressions Standalone expressions in most cases are meaningless. For example is not the product rule. This is just an expression that can evaluate to anything. However, if you associate the value of this expression with a value of another expression, like in , you know that you can substitute one for the other. Then if you want to know , which is in most cases impossible to calculate directly, you can instead calculate . A bunch of expressions scattered all over a piece of paper also do not mean anything. But if you have something like

(start exp)=exp2=exp3=…=expn=(end exp), because every expression equals the previous, you can conclude that

(start exp)=(end exp). You must use the equality sign and thus show how you get your solution.

Mysterious variables If you introduce a new variable, you must say what that new variable represents. If you do not do so, there is no way of knowing whether your solution is correct. The new mystery variable can take any value since its meaning is not defined, and thus if you solution uses that variable, it can also be very ambiguous.

Because

If the equality would hold in the red expressions, it would have to be that

Thus f and g would have to be very special functions. And those equalities would be useless to us.

Friendly operators It is not possible to have two operators next to each other. An operator can only stand between two variables or constants. For example, or are not valid expressions. or are.

coke ≠ vending machine A coke and a vending machine are not the same. A vending machine takes money as input and gives a can of coke as output. Depending on how much money you put in, you can get different thing out, according to their price. However, you can neither get a vending machine for a dollar nor can you take a vending machine home and leave a can of coke in its place. Functions and constants are pretty much the same. A function is like a vending machine. It takes constants as input and gives constants as output. Depending on what number it gets as input, it outputs a different number according to its definition. But a function is not a number. Let . You can give it different numbers as input.

, and get different numbers as output. One cannot, however, write, because is a vending machine and is a can of coke that costs $1 and comes out of the machine when we put in one dollar. Using this metaphor, the derivative of any coke is 0. The derivative of a vending machine is another vending machine, with the prices changed. Thus

. If you start with

, and then differentiate

you get another vending machine. But if you now want to put in a coin, you have to start by putting in a coin.

. You cannot do this

nor this

The last two are equating a coke with a vending machine. Think about it.