Implicit differentiation "in terms of $x$ and $y$". I dont get it when it means "to find $dy/dx$ in terms of $x$ and $y$" what does it mean?
Solve the following problem and write your answer and solutions on bond paper. 
1) Use implicit differentiation to find $dy/dx$ in terms of $x$ and $y$. 
a) $2x^3=2y^2+5$
b) $1=3x+2x^2y^2$
c)$x^2=(4x^2y^3+1)^2$
2) Use implicit differentiation to find second derivative in terms of $x$ and $y$. 
$4y^2+2=3x^2$
 A: It means that in your answer you are required to have only x and y as independent variables*, and not $y'$ or $\frac{dy}{dx}$ or anything else.
As far as I understand, with the first derivative it can't be done other way, since you don't introduce any other variables into the equation, like here, from your example:
$$
2x^2=2y^2+5 \to 4x = 4yy' \to y' = \frac{x}{y}
$$
Here the answer contains only x and y, which means exactly "in terms of x and y".
When one needs the second derivative, this is where you can have another variable, namely the first derivative. So you are required not to use it in your final answer, which usually means to substitute it with the first derivative equation (which contains only x and y)
From your question: Use implicit differentiation to find second derivative in terms of x and y:
$$
4y^2+2=3x^2 \\
8yy' = 6x \\
4yy' = 3x \\
y' = \frac{3x}{4y}
$$
this is the first derivative.
The second one is:
$$
4yy' = 3x \\
4y'y' + 4yy''=3 \\
4yy'' = 3 - 4(y')^2\\
y'' = \frac{3 - 4(y')^2}{4y}
$$
This is the second derivative, but it's not "in terms of x and y" since it contains $y'$ as an independent variable. You just need to substitute it with the value from $y' = \frac{3x}{4y}$, that is:
$$
y'' = \frac{3 - 4(\frac{3x}{4y})^2}{4y}\\
y'' = \frac{3}{4y} - \frac{9x^2}{16y^3}\\
$$
Here the equation contains only x and y as independent variables, which means it is in terms of x and y.

* and independent variable is one, which can be freely substituted for any allowed value, unlike the dependent one, which must be calculated from values of the independent variables.
