Use chain rule or implicit differentiation when solving the classic sliding ladder?

[rayroshi]

Would somebody please explain which it is that's used, when solving a classic "sliding ladder" related rates problem? I understand how to set up the diagram, label everything, and how to go through all of the steps, getting the solution, but I don't really understand what it is that we are using (chain rule or implicit differentiation?) and, more importantly, why it is used. In other words, I can get the right answer easily and every time, but I don't really understand why I am differentiating a^2 + b^2 = c^2 as (2a)(da/dt) + (2b)(db/dt) = (2c)(dc/dt). Is that implicit differentiation or is it using the chain rule? Instead of just going through the steps mindlessly, I would like to know the reason for those steps, of course. Trying to learn calculus on your own isn't easy, I've found, especially with a 73-year-old brain, lol!

All of the YouTube videos simply take you through the steps, but at that crucial point where you have to differentiate, they just do it, without explaining why that particular type of differentiation is used.

Any help would be much appreciated.

 

[stapel]

I don't really understand why I am differentiating a^2 + b^2 = c^2 as (2a)(da/dt) + (2b)(db/dt) = (2c)(dc/dt). Is that implicit differentiation or is it using the chain rule?

What is the definition of "the Chain Rule"? What is the definition of "implicit differentiation"? Is "a^2 + b^2 = c^2" a composition of functions, to which you are applying the former? Is it an equation in which you are differentiating each term with respect to some other, unstated (and thus "implicit") variable?

 

[markraz]

Would somebody please explain which it is that's used, when solving a classic "sliding ladder" related rates problem? I understand how to set up the diagram, label everything, and how to go through all of the steps, getting the solution, but I don't really understand what it is that we are using (chain rule or implicit differentiation?) and, more importantly, why it is used. In other words, I can get the right answer easily and every time, but I don't really understand why I am differentiating a^2 + b^2 = c^2 as (2a)(da/dt) + (2b)(db/dt) = (2c)(dc/dt). Is that implicit differentiation or is it using the chain rule? Instead of just going through the steps mindlessly, I would like to know the reason for those steps, of course. Trying to learn calculus on your own isn't easy, I've found, especially with a 73-year-old brain, lol!

All of the YouTube videos simply take you through the steps, but at that crucial point where you have to differentiate, they just do it, without explaining why that particular type of differentiation is used.

Any help would be much appreciated.

Which ladder problem? There are several approaches depending on what you want to find as it's moving. Are you trying to find the rate of the base? or the rate at the top as the ladder is sliding down? or is it the rate of angle? changing? or the area that you wish to find as the ladder is sliding? Here is a table of the common formulas.

problem

move

find

snap

formula

derivative

solve

have

need

ladder	+x	area	x	1/2xy	1/2x[y]+1/2y[x]	1/2x[v]+1/2y[rt]	dh	da
ladder	+x	-y	y	x^2+y^2=hy	2x[]+2y[]=0	x(rt)+y[]=0	dx	dy
ladder	+x	-y	t=c	x^2+y^2=hy	x[]+y[]=0	x(rt)+y[]=0	dx	dve
ladder	-y	+x	y	x^2+y^2=hy	x[]+y[]=0	x[]+y[-rt]=0	dy
ladder	-y	-x	y	x^2+y^2=hy	x[]+y[]=0	x[]+y[rt]=0	dy	dx
ladder	-x	+y	x	x^2+y^2=hy	x[]+y[]=0	x(rt)+y[]=0	dx	dy
ladder	+x	-y	t=c	x^2+y^2=hy	x[]+y[]=0	x(rt)+y[]=0	dx	dve
ladder	+x	ang f	y	cos(t)=1/hy	(1/hy)(rt)=-sin[]	((1/hy)(rt))/(y/hp)=[]	dx	dth
ladder	+x	ang w	th	sin=1/hy	x=o/h	(1/hy)(rt)=cos(x)[]	dx	dth
ladder	+x	ang w	th	sin=1/hy	cos=1/hy	(1/hy)(rt)=cos(x)[]	dx	dth

 

[rayroshi]

Which ladder problem?

Which ladder problem? There are several approaches depending on what you want to find as it's moving. Are you trying to find the rate of the base? or the rate at the top as the ladder is sliding down? or is it the rate of angle? changing? or the area that you wish to find as the ladder is sliding? Here is a table of the common formulas.

problem

move

find

snap

formula

derivative

solve

have

need

ladder	+x	area	x	1/2xy	1/2x[y]+1/2y[x]	1/2x[v]+1/2y[rt]	dh	da
ladder	+x	-y	y	x^2+y^2=hy	2x[]+2y[]=0	x(rt)+y[]=0	dx	dy
ladder	+x	-y	t=c	x^2+y^2=hy	x[]+y[]=0	x(rt)+y[]=0	dx	dve
ladder	-y	+x	y	x^2+y^2=hy	x[]+y[]=0	x[]+y[-rt]=0	dy
ladder	-y	-x	y	x^2+y^2=hy	x[]+y[]=0	x[]+y[rt]=0	dy	dx
ladder	-x	+y	x	x^2+y^2=hy	x[]+y[]=0	x(rt)+y[]=0	dx	dy
ladder	+x	-y	t=c	x^2+y^2=hy	x[]+y[]=0	x(rt)+y[]=0	dx	dve
ladder	+x	ang f	y	cos(t)=1/hy	(1/hy)(rt)=-sin[]	((1/hy)(rt))/(y/hp)=[]	dx	dth
ladder	+x	ang w	th	sin=1/hy	x=o/h	(1/hy)(rt)=cos(x)[]	dx	dth
ladder	+x	ang w	th	sin=1/hy	cos=1/hy	(1/hy)(rt)=cos(x)[]	dx	dth

I believe that I already showed which ladder problem, when I indicated the derivatives to be calculated; i.e., second one in your list: 2x + 2y...

 

[rayroshi]

What is the definition of "the Chain Rule"? What is the definition of "implicit differentiation"? Is "a^2 + b^2 = c^2" a composition of functions, to which you are applying the former? Is it an equation in which you are differentiating each term with respect to some other, unstated (and thus "implicit") variable?

Hi Stapel. Thanks for the answer. I like your approach of asking questions, trying to get someone to think, rather than just handing out answers; that's definitely the best way to teach. However, I'm barely past the chapter on differentiating in my "text," which is "Calculus for Dummies," a title which also gives you an idea of my math level. So I'm going to need a bit of hand holding and spoon feeding, here, since the ideas of implicit differentiation and the chain rule are still barely past the embryonic stage in my mind; still kinda fuzzy. From your prompts, it would seem that your second question fits best, when trying to find the instantaneous rate at which the top of the ladder is sliding down the wall (dy/dt), given the rate at which the base of the ladder is pulled away from the wall. I'm guessing that's true, because the equation y^2 + b^2 = length of ladder doesn't have time (t) anywhere, so it's an "implicit" variable. Is that right or wrong-headed thinking?Once again, thanks for your time and effort.

 

[stapel]

From your prompts, it would seem that your second question fits best, when trying to find the instantaneous rate at which the top of the ladder is sliding down the wall (dy/dt), given the rate at which the base of the ladder is pulled away from the wall. I'm guessing that's true, because the equation y^2 + b^2 = length of ladder doesn't have time (t) anywhere, so it's an "implicit" variable. Is that right or wrong-headed thinking?Once again, thanks for your time and effort.

The point of the Chain Rule is to show you how to differentiate in one context. The point of implicit differentiation is to explain differentiation in another context. Either way, you're differentiating. It's just that, depending on what you're differentiating, you'll use different methods or rules. That's all there is to it.