Is there an order of operations for the chain rule?
No, there is no specific order of operations when you are working problems using the chain rule. However, there is a key method to remember in order to keep your work straight as you accumulate work throughout the problem. The key method to remember is to work from the outside in. If you work from the outside in you can cross the components of the problem out as you complete them.
Why or Why not?
There is not order of operations because almost every single problem is different from each other. Some problems call for the product rule, quotient rule, simple derivatives, etc.. or any combination of the methods for derivatives used thus far in calculus. Because so many different combinations can be needed for each problem, one set of steps cannot be used because each component of each combination is solved differently. Keeping in mind to work from the outside in can serve as a starting point, however, the details of working from the outside in are relevant to each specific problem because a problem is able to have multiple components on the inside and outside.
Give several examples of chain rule problems.
Example 1: sqrt 3x^2-x+1 (3x^2-x+1)^1/2 ½ ((3x^2-x+1)^ -1/2 * (6x-1) 6x-1/ 2(3x^2-x+1)^1/2
Example 2: y= tan^2x 2tanx * (sec^2 x) 2tanxsec^2x
Example 3: (x^2+1)^3 3(x^2+1) ^2 * (2x) 6x (x^2+1)^2
Example 4: (3x-2x^2)^3 3(3x-2x^2)^2 * (3-4x) 3(3-4x)(3x-2x^2)^2
Example 5: cubert (x^2-1)^2 (x^2-1)^2/3 2/3(x^2-1)^-1/3 * (2x) 2(2x)/3(x^2-1)^1/3
Is there an order of operations for the chain rule?
-No, there is no order of operations for the chain rule. The chain rule could be just a chain rule, or a product rule and quotient rule combined into the chain rule. Therefore when working the problems you have to follow the formula pretty much. It is easiest to work the chain rule from the outside in.
Why or why not?
-There is no order of operations because each problem is different. Some may be very complicated and others simple. Some may also have more terms then other, some dealing with trig functions, and some may just have a constant and exponent. Keep in mind when working the chain rule: Derivative of the outside, Recopy the inside, Multiply by the derivative of the inside.
No, there is not technically an order of operations when using the chain rule. However, there are different ways in which you have to use it. For instance, you use it when something is in parenthesis raised to an exponent; you use it usually when something is under a square root; you can even use it at the same time as you may be doing the product or quotient rule. Here are some examples:
1.) square root of (3x+6) *First thing you have to do is change the squareroot to exponential form *So you get (3x+6)^1/2 *Now you can start the chain rule..First bring the 1/2 to the front and subtract 1 from 1/2 for the new exponent, so you should have: 1/2(3x+6)^-1/2 *now you multiply that^ by the derivative of the inside (what's in parenthesis). And that derivative would be 3 *So you have 1/2(3x+6)^-1/2(3) Simplifying that you get: (3)/2(3x+6)^1/2 which is the same as: (3)/2sqrt of 3x+6
2.) 3(-2x+3)^2 *what's different about this one is that there's a number out in front...So what should we do with that number?...Welllll you don't take the derivative. *You leave the 3 out in front, then bring the exponent, 2, out in front, recopy the inside, and subtract 1 from 2 (the exponent)..So you get: 3(2(-2x+3)) *Now you multiply that^ by the derivative of what's on the inside of the parenthesis, which is -2 *So you have 3(2(-2x+3))(-2) *And that gives you -12(-2x+3)
3.) y = 2cos(5x) *You know you have to use the chain rule for this one because there's something on the inside of the trig function, 5x. *So first, you leave the 2 out in front. Then take the derivative of cos, which is -sin, and recopy the 5x so it should look like this: 2(-sin(5x)) *Now muptiply that by the derivative of what's left (5x), and that derivative would be 5 *So you have 2(-sin(5x))(5) *Simplifying that you get -10sin5x
No, there is not any specific order of operations for the chain rule. BUT, there is a certain way it needs to be done. Every problem is different, so there can’t be a set way to work a chain rule. For some problems, the chain rule is on the inside (ex. x/((x+2)^3)) and others, the chain rule is on the outside (ex. (x/x+1)^2). Whatever is on the outside, whether it be the chain rule or a quotient rule or a product rule, that’s what you focus on first.
Ex. 1) y = x/((x+2)^3) For this problem, the chain rule is on the inside, so you focus on the quotient rule first. y’ = ((x + 2)^3)(1) – (x)(3(x + 2)^2 (1)) Notice how I used the quotient rule (bottom times derivative of top minus top times derivative of bottom all over bottom squared), but when I got to the derivative of the bottom, I used the chain rule. y’ = (x + 2)^3 – 3x(x + 2)^2 *Remember: simplify your answer as much as possible.
Ex. 2) f(x) = (x/x + 1)^2 For this problem, the chain rule is on the outside, so you focus on the chain rule first. f ’(x) = 2(x/x + 1) (((x+1)(1) – (x)(1)) / (x + 1)^2) Notice how I used the chain rule (bring the exponent to the front, recopy the inside, minus one from the exponent, then multiply by the derivative of the inside), but when I got to the derivative of the inside, I used the quotient rule. f ’(x) = 2(x / x + 1) (1 / (x + 1)^2) f ’(x) = 2(x / (x + 1)^3)
There is no certain order of operations that is used when using the chain rule but there is one way it has to be done. you ALWAYS work the outside in ex (x^2+5x)^6 or |x^2+3 |=sqrt.
1.(x^2+5x)^6 The 6th power is on the outside, so you bring the exponent to the front, subtract 1 from the exponent outside, then multiply by the derivative of the inside. 6(x^2+5x)^5 (2x+5) 12x+30(x^2+5x)^5
2.sin(cos(x)) sin is the outer part of this problem, take the derivative of sin, recopy inside, multiply by the derivative of the inside. cos(cos(x))(-sinx) -sinxcos(cos(x))
There is an order of operations that is used when you take the derivative by using the chain rule sometimes when you use calculus. There is an order of operations because you have to do certain things to "take" the derivative of a function, as in with a parentheses raised to an exponent, or different roots. I'll maybe give a few examples as stated above, but I might just give one for the fuck of it, for the lawls.
Find the derivative by using the chain rule.
the square root of (2x^2 - 7) (2x^2 - 7)^1/2 = 1/2(2x^2 - 7)^-1/2 x (4x) 2x(2x^2 - 7)^-1/2 (2x)/((2x^2 - 7)^1/2)
i dont think there is an order of operations when you are using chain rule because of all the different equations that it can be applied to but there could be if you simply eliminate steps as they do not apply. chain rule seems to apply to trig functions most often.
how can there be an answer to "Why or why not?" that's like asking "Is the pizza ready?" and asking "How does it feel about being eaten?" right after that
I think that there is. You have to use the order of operations to complete the chain rule. If you multiplied before you raised it by an exponent. There are things that you must do with the order of operations but its not a big role really in the chain rule. The chain rule is its own thing and the order or operations just kind of plays into it. P.S. Curtis just murked us. ha
This comment has been removed by a blog administrator.
ReplyDeleteIs there an order of operations for the chain rule?
ReplyDeleteNo, there is no specific order of operations when you are working problems using the chain rule. However, there is a key method to remember in order to keep your work straight as you accumulate work throughout the problem. The key method to remember is to work from the outside in. If you work from the outside in you can cross the components of the problem out as you complete them.
Why or Why not?
There is not order of operations because almost every single problem is different from each other. Some problems call for the product rule, quotient rule, simple derivatives, etc.. or any combination of the methods for derivatives used thus far in calculus. Because so many different combinations can be needed for each problem, one set of steps cannot be used because each component of each combination is solved differently. Keeping in mind to work from the outside in can serve as a starting point, however, the details of working from the outside in are relevant to each specific problem because a problem is able to have multiple components on the inside and outside.
Give several examples of chain rule problems.
Example 1: sqrt 3x^2-x+1
(3x^2-x+1)^1/2
½ ((3x^2-x+1)^ -1/2 * (6x-1)
6x-1/ 2(3x^2-x+1)^1/2
Example 2: y= tan^2x
2tanx * (sec^2 x)
2tanxsec^2x
Example 3: (x^2+1)^3
3(x^2+1) ^2 * (2x)
6x (x^2+1)^2
Example 4: (3x-2x^2)^3
3(3x-2x^2)^2 * (3-4x)
3(3-4x)(3x-2x^2)^2
Example 5: cubert (x^2-1)^2
(x^2-1)^2/3
2/3(x^2-1)^-1/3 * (2x)
2(2x)/3(x^2-1)^1/3
Example 6: -7/(2x-3)^2
(2x-3)^2 (0) – [-7(2(2x-3) * (2))]/( 2x – 3^2)^2
-[-28 (2x-3)]/(2x-3)^4
28(2x-3)/2(-3)^4
28/(2x-3)^3
Example 7: x/ cubert x^2 + 4
x/ (x^2+4)^1/3
(x^2+4)^1/3 * (1) – [x(1/3x^2 +4) ^-2/3 * (2x)]/ (x^2+4)^2/3
(x^2+4)^1/3-[2/3x^2(x^2+4)^-2/3]/ (2x)]/ (x^2+4)^2/3
(x^2+4)^-2/3[ (x^2+4)- 2/3 x^2]/ (x^2+4)^2/3
X^2+4-2/3x^2/ (x^2+4)^4/3
1/3x^2+4/ (x^2+4)^4/3
Example 8: (3x-1/x^2+3)^2
2(3x-1/x^2 +3) * (x^2+3(2)-[3x-1(2x)]/(x^2+3)^2)
2(3x-1/x^2+3) (9x^2+9-6x^2+2x/(x^2+3)^2)
2((3x-1)(3x^2+2x+9)/(x^2+3)^2)
Example 9: sin (2x)
Cos(2x) * (2)
2 cos (2x)
Example 10: sin ^3 4t
3(sin^2 4t) * (cos 4t) * (4)
12 ( sin^2 4t)(cos4t)
6 sin 8t sin 4t
Is there an order of operations for the chain rule?
ReplyDelete-No, there is no order of operations for the chain rule. The chain rule could be just a chain rule, or a product rule and quotient rule combined into the chain rule. Therefore when working the problems you have to follow the formula pretty much. It is easiest to work the chain rule from the outside in.
Why or why not?
-There is no order of operations because each problem is different. Some may be very complicated and others simple. Some may also have more terms then other, some dealing with trig functions, and some may just have a constant and exponent. Keep in mind when working the chain rule: Derivative of the outside, Recopy the inside, Multiply by the derivative of the inside.
Examples:
x/ cube root x^2 + 4 x/ (x^2+4)^1/3 (x^2+4)^1/3 * (1) – [x(1/3x^2 +4) ^-2/3 * (2x)]/ (x^2+4)^2/3 (x^2+4)^1/3-[2/3x^2(x^2+4)^-2/3]/ (2x)]/ (x^2+4)^2/3 (x^2+4)^-2/3[ (x^2+4)- 2/3 x^2]/ (x^2+4)^2/3 X^2+4-2/3x^2/ (x^2+4)^4/3 1/3x^2+4/ (x^2+4)^4/3
(x^2+3)^3
Outer:y=u^3
Inner:u=X^2+3
3(x^2+3)(2x)
3(2x)(x^2+3)
3x + 2tan(3x)^2
3 + (4sec^2(3x))(18x)
3 + 72sec^2(3x)
No, there is not technically an order of operations when using the chain rule. However, there are different ways in which you have to use it. For instance, you use it when something is in parenthesis raised to an exponent; you use it usually when something is under a square root; you can even use it at the same time as you may be doing the product or quotient rule.
ReplyDeleteHere are some examples:
1.) square root of (3x+6)
*First thing you have to do is change the squareroot to exponential form
*So you get (3x+6)^1/2
*Now you can start the chain rule..First bring the 1/2 to the front and subtract 1 from 1/2 for the new exponent, so you should have:
1/2(3x+6)^-1/2
*now you multiply that^ by the derivative of the inside (what's in parenthesis). And that derivative would be 3
*So you have 1/2(3x+6)^-1/2(3)
Simplifying that you get:
(3)/2(3x+6)^1/2 which is the same as:
(3)/2sqrt of 3x+6
2.) 3(-2x+3)^2
*what's different about this one is that there's a number out in front...So what should we do with that number?...Welllll you don't take the derivative.
*You leave the 3 out in front, then bring the exponent, 2, out in front, recopy the inside, and subtract 1 from 2 (the exponent)..So you get:
3(2(-2x+3))
*Now you multiply that^ by the derivative of what's on the inside of the parenthesis, which is -2
*So you have 3(2(-2x+3))(-2)
*And that gives you -12(-2x+3)
3.) y = 2cos(5x)
*You know you have to use the chain rule for this one because there's something on the inside of the trig function, 5x.
*So first, you leave the 2 out in front. Then take the derivative of cos, which is -sin, and recopy the 5x so it should look like this:
2(-sin(5x))
*Now muptiply that by the derivative of what's left (5x), and that derivative would be 5
*So you have 2(-sin(5x))(5)
*Simplifying that you get -10sin5x
This comment has been removed by the author.
ReplyDeleteNo, there is not any specific order of operations for the chain rule. BUT, there is a certain way it needs to be done. Every problem is different, so there can’t be a set way to work a chain rule. For some problems, the chain rule is on the inside (ex. x/((x+2)^3)) and others, the chain rule is on the outside (ex. (x/x+1)^2). Whatever is on the outside, whether it be the chain rule or a quotient rule or a product rule, that’s what you focus on first.
ReplyDeleteEx. 1) y = x/((x+2)^3)
For this problem, the chain rule is on the inside, so you focus on the quotient rule first.
y’ = ((x + 2)^3)(1) – (x)(3(x + 2)^2 (1))
Notice how I used the quotient rule (bottom times derivative of top minus top times derivative of bottom all over bottom squared), but when I got to the derivative of the bottom, I used the chain rule.
y’ = (x + 2)^3 – 3x(x + 2)^2
*Remember: simplify your answer as much as possible.
Ex. 2) f(x) = (x/x + 1)^2
For this problem, the chain rule is on the outside, so you focus on the chain rule first.
f ’(x) = 2(x/x + 1) (((x+1)(1) – (x)(1)) / (x + 1)^2)
Notice how I used the chain rule (bring the exponent to the front, recopy the inside, minus one from the exponent, then multiply by the derivative of the inside), but when I got to the derivative of the inside, I used the quotient rule.
f ’(x) = 2(x / x + 1) (1 / (x + 1)^2)
f ’(x) = 2(x / (x + 1)^3)
There is no certain order of operations that is used when using the chain rule but there is one way it has to be done. you ALWAYS work the outside in ex (x^2+5x)^6 or |x^2+3 |=sqrt.
ReplyDelete1.(x^2+5x)^6
The 6th power is on the outside, so you bring the exponent to the front, subtract 1 from the exponent outside, then multiply by the derivative of the inside.
6(x^2+5x)^5 (2x+5)
12x+30(x^2+5x)^5
2.sin(cos(x))
sin is the outer part of this problem, take the derivative of sin, recopy inside, multiply by the derivative of the inside.
cos(cos(x))(-sinx)
-sinxcos(cos(x))
3.(x+4)^-3
-3(x+4)^-4(1)
-3/(x+4)^4
There is an order of operations that is used when you take the derivative by using the chain rule sometimes when you use calculus. There is an order of operations because you have to do certain things to "take" the derivative of a function, as in with a parentheses raised to an exponent, or different roots. I'll maybe give a few examples as stated above, but I might just give one for the fuck of it, for the lawls.
ReplyDeleteFind the derivative by using the chain rule.
the square root of (2x^2 - 7)
(2x^2 - 7)^1/2 = 1/2(2x^2 - 7)^-1/2 x (4x)
2x(2x^2 - 7)^-1/2
(2x)/((2x^2 - 7)^1/2)
there you go.
i dont think there is an order of operations when you are using chain rule because of all the different equations that it can be applied to but there could be if you simply eliminate steps as they do not apply. chain rule seems to apply to trig functions most often.
ReplyDeletefor example: sin^3(x)
3sin^2(x)(cosx)
3cos(x)sin^2(x)
is there an order of operations to a chain rule?
ReplyDeleteI think there is
how can there be an answer to "Why or why not?"
that's like asking "Is the pizza ready?" and asking "How does it feel about being eaten?" right after that
examples....
(6x^2)^2
2(6x^2)*12x
12x^2*12x
144x^3
(4x^5)^3
3(4x^5)^2*20x^4
48x^10*20x^4
96x^14
I think that there is. You have to use the order of operations to complete the chain rule. If you multiplied before you raised it by an exponent. There are things that you must do with the order of operations but its not a big role really in the chain rule. The chain rule is its own thing and the order or operations just kind of plays into it. P.S. Curtis just murked us. ha
ReplyDelete