a)
b)
c) The two expressions match
Answer:
a)
The equation in this problem is
Here, we want to find by implicit differentiation.
To do that, we apply the operator on each term of the equation. We have:
(by applying composite function rule)
Therefore, the equation becomes:
And re-arranging for dy/dx, we get:
b)
Now we want to solve the equation explicitly for y and then differentiate to find dy/dx. The equation is:
First, we isolate y, and we find:
And taking the square root,
Here we are told to consider only the first and second quadrants, so those where y > 0; so we only take the positive root:
Now we differentiate this function to find dy/dx; using the chain rule, we get:
(2)
c)
Now we want to check if the two solutions are consistent.
To do that, we substitute the expression that we found for y in part b:
Into the solution found in part a:
Doing so, we find:
(1)
We observe that expression (1) matches with expression (2) found in part b: therefore, we can conclude that the two solutions are coeherent with each other.
