This is a problem from my practice midterm. No answers are given. Can someone please tell me if I have answered this problem correctly?
S=Distance between P & Airplane
I take the derivative of & S with respect to
Is my work correct?
Is it OK that I only took the derivative of S (with respect to theta) on the denominator only on the right hand side, or should I have taken the derivative of the entire right hand side using the quotient rule?
In my practice problems, when doing implicit derivatives, we take the derivatives of both sides in their entirety. However, I am watching a youtube video on related rates regarding the volume of a balloon, and the guy only takes the derivatives of the two variables he is interested in. Can someone please elaborate on this?
Thank you!
Thanks. I'm not quite sure I understand what you did here. First of all, you solved for "S".
I thought with implicit differentiation, you should not do this, and solve for dy/dx at the very end?
Ok, then you differentiated the left hand side. Makes sense. What exactly are you doing to the right hand side? This isnt the quotient rule is it?
Are we not taking the deriv of the ENTIRE side, but JUST the variables we are interested in, correct? In this case it was ?
Why couldnt you have just done: ?
By doing it that way, you are not correctly differentiating the right-hand-side.
The question asks "At what rate is the distance s between the airplane and the fixed point P changing with when ."
This is asking for evaluated for
Hence, we write in terms of and differentiate.
One option is to use the quotient rule for "either" of the following equivalent expressions.
Alternatively, you can differentiate one of the alternative expressions.
But you must differentiate the entire expression of the RHS, not just the denominator.
Ok, I think I follow you.
so we have
and we have to take the derivative of the entire RHS. But, since we are only interested in the
part of it, we are allowed to pull the 500 outside the derivative expression.
So, now we are allowed to treat as one term which we can take the derivative of, correct?
You stopped at:
I thought the derivative of ?
But,
couldnt you go further?
and the final answer is