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.
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 (
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.