and then multiply that by .
4. The derivative of x3 is 3x2, so you have
5. To simplify, rewrite the sine power and move the 3x2 to the front.
That’s a wrap.
With chain rule problems, never use more than one derivative rule per step. In other words, when you do the derivative rule for the outermost function, don’t touch the inside stuff! Only in the next step do you multiply the outside derivative by the derivative of the inside stuff.
17 18
19 20
*21
*22
*23
*24
25
26
27 What’s at the point ?
28
What to Do with Y’s: Implicit Differentiation
You use implicit differentiation when your equation isn’t in “ ” form, such as , and it’s impossible to solve for y. If you can solve for y, implicit differentiation will still work, but it’s not necessary.
Implicit differentiation problems are chain rule problems in disguise.
Here’s what I mean. You know that the derivative of is , and that according to the chain rule, the derivative of is
. You would finish that problem by doing the derivative of , but I have a reason for leaving the problem unfinished here.
To do implicit differentiation, all you do (sort of) is every time you see a
“y” in a problem, you treat it like the is treated above. Thus, because the derivative of is , the derivative of is . Then, after doing the differentiation, you just solve for so that
you get .
By the way, I used “y” in the preceding explanation, but that’s not the whole story. Consider that is the same as . It’s the variable on the top that you apply implicit differentiation to. This is typically y, but it could be any other variable. And it’s the variable on the bottom that you treat the ordinary way. This is typically x, but it could also be any other variable.
Q. If , find .
A. .
1. Take the derivative of all four terms, using the chain rule for terms containing y and using the ordinary method for terms containing x.
2. Move all terms containing to the left side and all other terms to the right side.
3. Factor out . 4. Divide.
That’s your answer. Note that this derivative — unlike ordinary derivatives — contains y’s as well as x’s.
29 If , find by implicit differentiation.
30 If , find .
31 For , find by implicit differentiation.
*32 If , find the slope of the curve at .
33 If find .
34 Find the slope of the line tangent to the circle at the point .
35 If , find .
36 Find the slope of the normal line to the ellipse at the
point .
Getting High on Calculus: Higher Order Derivatives
You often need to take the derivative of a derivative, or the derivative of a derivative of a derivative, and so on. In the next two chapters, you see a few applications. For example, a second derivative tells you the acceleration of a moving body. To find a higher order derivative, you just treat the first derivative as a new function and take its derivative in the ordinary way. You can keep doing this indefinitely.
37 For , find the 1st through 6th derivatives. Extra credit: What’s the 2,015th derivative?
38 For , find the 1st, 2nd, 3rd, and 4th derivatives.
39 For , find the 1st through 6th derivatives.
40 For , find the 1st, 2nd, and 3rd derivatives.
41 For , find the 6th derivative.
*42 For , find the 4th derivative.
Solutions for Differentiation Problems
1 ; .
The derivative of any constant is zero.
2 ; .
Don’t forget that even though sort of looks like a variable (and even though other Greek letters like θ, α, and ω are variables), is a number (roughly 3.14) and behaves like any other number. The same is true of . And when doing derivatives, constants like c and k also behave like ordinary numbers.
Because is just a number, is also just a number. is, therefore, a horizontal line with a slope and a derivative of zero.
3 (where k is a constant); .
If you feel bored because the first few problems were so easy, just enjoy it; it won’t last.
4 ; .
Bring the 4 in front and multiply it by the 5, and at the same time reduce the power by 1, from 4 to 3: . Notice that the coefficient 5 has no effect on how you do the derivative in the following sense: You could ignore the 5 temporarily, do the derivative of x4 (which is 4x3), and then put the 5 back where it was and multiply it by 4.
5 ;
You can just write down the derivative without showing any work (bring the 3 in front of the x, reduce the power 3 to a 2, and the 10 sits there doing nothing):
But if you want to do it more methodically, it works like this:
1. Rewrite so you can see an ordinary coefficient: . 2. Bring the 3 in front, multiply, and reduce the power by 1.
This is the same, of course, as .
6 ; .
Rewrite with an exponent and finish like Problem 5: Bring the power in front and reduce the power by one: .
To write your answer without a negative power, you write or . Or you can write your answer without a fraction power, to wit:
or or or . You say “po-tay-to”; I say
“po-tah-to.”
7 ; .
Note that the derivative of plain old t or plain old x (or any other variable) is simply 1. In a sense, this is the simplest of all derivative rules, not counting the derivative of a constant. Yet for some reason, many people get it wrong. This is simply an example of the power rule:
x is the same as x1, so you bring the 1 in front and reduce the power by 1, from 1 to 0. That gives you 1x0. But because anything to the 0 power equals 1, you have 1 times 1, which of course is 1.
8 ; .
FOIL and then take the derivative.
9 .
Remember that . For a great mnemonic to help you remember the derivatives of the other four trig functions, check out Chapter 17.
10 .
A helpful rule: .
11 .
Another helpful rule: .
When doing this derivative, you can deal with the “5” in two ways.
First, you can ignore it temporarily, do the differentiating, then multiply your answer by 5. (If you do it this way, don’t forget that the “5”
multiplies the entire derivative, not just the first term.) The second way is probably easier and better: Just make the “5” part of the first function.
To wit:
*12 .
This is a challenge problem because, as you’ve probably noticed, there are three functions in this product instead of two. But it’s a piece o’
cake. Just make it two functions: either or . Take your pick.
A handy rule: (Note that and its multiples [like ] are the only functions that are their own derivatives.)
1. Rewrite this “triple function” as the product of two functions.
2. Apply the product rule.