Why Is Implicit Differentiation Used Here?

[rayroshi]

In a problem involving related rates, the solution involves implicit differentiation, I have two questions

1. Why is implicit differentiation used?
2. When setting up the solution to implicit differentiation, what is the real reason that the additional term is tacked on?

The problem involves two cars which have left an intersection, one (car 'a') heading north at 50 m.p.h., and the other (car 'b') heading west at 30 m.p.h. The problem ask you to solve the rate at which the distance between the two cars (c) is changing at the moment when the first car is 0.3 mi. and the second car is 0.4 mi., north and south of the intersection, respectively.

Since the cars' velocities involve a right triangle, the Pythagorean formula is used, c^2 = a^2 + b^2,

And, given the fact that the problem is asking for a rate, we are then being asked to solve for a derivative, which, in this case, is dc/dt.

The solution is reached by differentiating the equation, as follows: (2c)(dc/dt) = (2a)(da/dt) + (2b)(db/dt), using the power rule for differentiation, as well as implicit differentiaton.

So, to my first question: Why the implicit differentiation? Is it because we are differentiating with respect to time, but there is no time shown in the original equation?

I am just learning implicit differentiation, so I'm still not real sure about when it is applicable. I understand that, as the name implies, it is used when an equation doesn't explicitly show the relationship between two variables, as in an equation such as y = 3x, but I'm still not really sure about the underlying rationale for how and when to use it.

In Mark Ryan's book, Calculus for Dummies, p.147, second paragraph, he talks about being able to "think of 'y' as being equal to some unknown mystery function of 'x'." However, that has no meaning to me. And, in her YouTube video on implicit differentiation, Nancy Pi (

) says, at 3 minutes into the video, that you need to tack on the extra terms (such as the dc/dt, da/dt, db/dt, in the above problem, for example) because the original terms being differentiated are being treated as though using the chain rule, with the outside function being the coefficient & the power, and the inside term being the variable, itself, which she says "the 'y' contains some 'x' expression." I have no idea what that really means. I guess it is this last phrase (the inside term being the variable, itself, which she says "the 'y' contains some 'x' expression") that I would really like to understand.

Using phrases like, 'some mystery function' or 'contains some x function' leave me wondering just what was meant, and it is this that I would like to learn.

Any help would be greatly appreciated.

 

[skeeter]

So, to my first question: Why the implicit differentiation? Is it because we are differentiating with respect to time, but there is no time shown in the original equation?

yes, the variables representing the distances of both cars are implicitly dependent on time

The problem involves two cars which have left an intersection, one (car 'a') heading north at 50 m.p.h., and the other (car 'b') heading west at 30 m.p.h. The problem ask you to solve the rate at which the distance between the two cars (c) is changing at the moment when the first car is 0.3 mi. and the second car is 0.4 mi., north and south of the intersection, respectively.

neither car will be south of the intersection if their respective directions are north and west

further, car "a" will not be 0.3 miles north of the intersection at the same time car "b" is 0.4 miles west of the intersection.

get the problem statement fixed, and one can set up the motion equations explicitly as functions of time if so desired

 

[Dr.Peterson]

neither car will be south of the intersection if their respective directions are north and west

further, car "a" will not be 0.3 miles north of the intersection at the same time car "b" is 0.4 miles west of the intersection.

I agree the second car must be west of the intersection, which is probably a typo. But it doesn't say they left at the same time, so it's possible for them to be 0.3 and 0.4 miles away at the indicated time.

1. Why is implicit differentiation used?
2. When setting up the solution to implicit differentiation, what is the real reason that the additional term is tacked on?
...
So, to my first question: Why the implicit differentiation? Is it because we are differentiating with respect to time, but there is no time shown in the original equation?

I am just learning implicit differentiation, so I'm still not real sure about when it is applicable. I understand that, as the name implies, it is used when an equation doesn't explicitly show the relationship between two variables, as in an equation such as y = 3x, but I'm still not really sure about the underlying rationale for how and when to use it.

In general, implicit differentiation is used when it makes the work easier, or possible. In this case, you could solve for c and differentiate explicitly (keeping the chain rule in mind), but working with a radical would make the work harder. In other cases, you might not be able to solve for on variable at all, so only implicit differentiation can work.

There is no "additional term"; I suppose you mean the extra factor of a derivative. That's the chain rule; without it, you would be differentiating with respect to a, b, and c, not t.

In Mark Ryan's book, Calculus for Dummies, p.147, second paragraph, he talks about being able to "think of 'y' as being equal to some unknown mystery function of 'x'." However, that has no meaning to me. And, in her YouTube video on implicit differentiation, Nancy Pi says, at 3 minutes into the video, that you need to tack on the extra terms (such as the dc/dt, da/dt, db/dt, in the above problem, for example) because the original terms being differentiated are being treated as though using the chain rule, with the outside function being the coefficient & the power, and the inside term being the variable, itself, which she says "the 'y' contains some 'x' expression." I have no idea what that really means. I guess it is this last phrase (the inside term being the variable, itself, which she says "the 'y' contains some 'x' expression") that I would really like to understand.

Using phrases like, 'some mystery function' or 'contains some x function' leave me wondering just what was meant, and it is this that I would like to learn.

These sources are using such vague terms in an attempt to be informal and friendly! If they don't communicate to you, then you could just look for someone else who writes in a style that works for you.

There's really no "mystery"; it's just that y is a function of x, in the sense that it is dependent on the value of x. When a function depends on y, and y itself depends on x, then you need the chain rule to differentiate the entire function with respect to x.

And the "inside function" means the g in f(g(x)) in the full statement of the chain rule. The y in the larger expression is a function of x that is inside the whole expression. And (though her way of saying it doesn't work for me) I suppose she is imagining that if you knew how y is related to x (e.g. y = x^2), then you could replace the y with x^2, and that expression would be inside the larger expression.

 

[skeeter]

But it doesn't say they left at the same time, so it's possible for them to be 0.3 and 0.4 miles away at the indicated time.

Possible, yes, but still should be stated that the cars left at different times. Either way, an assumption is required.

 

[Dr.Peterson]

Possible, yes, but still should be stated that the cars left at different times. Either way, an assumption is required.

Not really. Just don't assume that they left at the same time, since the two specified distances contradict that assumption, and also make it unnecessary to know anything about when they left. Assume only what is stated.