Master AP Calculus AB & BC


CHAPTER 4. Differentiating


If asked to find the derivative of x, you would probably give an answer of 1. Technically, however, the derivative of x is dx. (Similarly, the derivative of y is dy, etc.) The derivative of x is only 1 when you are differentiating with respect to x. Examine more closely the notation dy/dx that you have unknowingly used all this time. This notation literally means “the derivative of y with respect to x.” The numerator contains the variable you are differentiating, and the denominator contains the variable you are “respecting.” Using this notation, the derivative of x with respect to x is dx/dx, or 1.

Let’s look at a more complicated example. You know that the derivative of x3 (with respect to x) is 3x2. Let’s be even more careful about the process and use the Chain Rule. The derivative of x3with respect to x is 3x2 times the derivative of what’s inside, x, with respect to x: dx/dx. The final answer is If the variable in the expression matches the variable you are “respecting,” you differentiate like you have in previous sections. When the variables don’t match, however, things get a little more bizarre.

TIP. Don’t forget: da/db is read “the derivative of a with respect to b." Use this notation a ≠ b.

Example 15: Find the derivative of 5y4 with respect to x.

Solution: Apply the Chain Rule to this expression: the y is the inner function, substituted into x4. Thus, take the derivative of the outer function leaving y alone,


and then multiply by the derivative of the inside (y) with respect to x (dy/dx). Your final answer is

It is very important to notice when you are differentiating with respect to variables that don’t match those in the expression. This only happens rarely, but it is important when it happens.

ALERT! No math joke is really that funny, so although the Smp sons joke is pretty amusing to us math geeks, you risk serious physical injury if you try to tell it in public.

The television show The Simpsons used this concept of differentiation in the episode where Bart cheats his way into a school for smart kids. The teacher asks the students to determine what is funny about the derivative of r3/3. The derivative is (technically) r2dr, or r ∙ dr ∙ r (hardy-har-har). In this problem, dr is the derivative of r because no respecting variable was defined. Had the teacher asked for the derivative of r3/3 with respect to r, the joke would have been less funny, since that derivative is r2.

Example 16: Find the derivative of each with respect to x:

(a) t

The derivative of t with respect to x is written dt/dx

NOTE. The skill of d differentiating variables with respect to other variables happens most often in implicit differentiation, related rates, and differential equations.

(b) 7x3 sin x

This expression contains two different functions, so you need to use the Product Rule: 7x3 cos x + 21x2 sin x. There is no fraction like the dt/dx from the last problem, because you are taking the derivative with respect to the variable in the expression (they match).

(c) cos y2

This is a composite function and will require the Chain Rule: The y is plugged into x2, which is plugged into cos x. The derivative, according to the Chain Rule, is —sin y2 2y ∙ dy/dx. Notice that you leave the y alone until the very end, as it is the innermost function.

(d) 2xy

This is the product of 2x and y, so use the Product Rule: The derivative of 2x with respect to x is simply 2.


Directions: Solve each of the following problems. Decide which is the best of the choices given and indicate your responses in the book.


1. What is meant by dg/dc? Create an expression whose derivative contains dg/dc.

2. Find the derivative of each with respect to y:

(a) csc 2y

(b) ex+y

(c) cos (ln x) + xy2


1. dg/dc is the derivative of g with respect toe. This expression appears whenever you differentiate an expression containing g’s with respect to c. For example, the derivative of g with respect to c (mathematically,

2. (a) —2csc 2y cot 2y: Don’t forget to leave the 2y alone as you find the derivative of cosecant; finally, multiply by the derivative of 2y with respect toy, which is 2.

(b) : Remember, you are looking for the derivative of x with respect toy in this problem.

(c) This problem requires the Chain Rule for the first term and the Product Rule for the second: