You do the implcit differentation, then solve for y' and check where this is defined.
In your case: Differentiate implicitly: 2xy + x²y' - y² - x*2yy' = 0
Solve for y': y'(x²-2xy) +2xy - y² = 0
y' = (2xy-y²) / (x²-2xy)
Check where defined: y' is not defined if the denominator becomes zero, i.e.
x² - 2xy = 0 x(x - 2y) = 0
This has formal solutions x=0 and y=x/2. Now we check whether these values are possible for the initially given definition of y:
0^2*y - 0*y^2 =? 4 0 =? 4
This is impossible, hence the function is not defined for 0, and we can disregard this.
x^2*(x/2) - x(x/2)^2 =? 4 x^3/2 - x^3/4 = 4 x^3/4 = 4 x^3=16 x^3 = 16 x = cubicroot(16)
This is a possible value for y, so we have a point where y is defined, but not y'.
The solution to all of it is hence D - { cubicroot(16) }, where D is the domain of y (which nobody has asked for in this example :-).
(Actually, the check whether 0 is in D is superfluous: If you write as solution D - { 0, cubicroot(16) }, this is also correct - only it so happens that 0 is not in D, so the set difference cannot take it out of there ...).
If someone asks for that D, you have to solve the definition for y and find that domain - I don't know of any [general] way to find the domain without solving for the explicit function).
It will be 52 as you know when looking at a triangle with 3 angles they must all add up to 180 so you simply do 56+72 then subtract 180 from it and you get 52.
The logarithm of a quotient can be written as a difference of logarithms:
You can also think of this as a combination of the product-to-sum and reciprocal/power properties of logarithms:
To summarize,
To find his age 4 years ago, you need to subtract 4 from his current age.
His current age is Y, so his age 4 years ago is: y - 4
If y equals 2x - 3 in the second equation, than subsitute it in for y in the first.
2x + 2x - 3 = 5
4x - 3 = 5
4x = 8
x=2
2(2) + y = 5
x = 2 and y = 1
