Confused about partial derivatives I am having some issues understanding what should I keep constant and what not in certain cases when I take partial derivatives. Specifically in this kind of situation: say we have a function
$$f(x,y) = x^3+7y^2$$
and we also know that $y=2x+1$ and we need to find the partial derivative with respect to $x$ at a given point, say $\frac{\partial f(2,3)}{\partial x}$. From what I understood, when taking partial derivatives with respect to a variable, you need to keep the other constant, in which case, if I do that above (keeping $y$ constant) I would get, $\frac{\partial f(x,y)}{\partial x}=3x^2$, so I get $12$. However, if I plug $x$ in $y$ explicitly I would get $$f(x,y) = x^3+7(2x+1)^2$$ so,
$$\frac{\partial f(x,y)}{\partial x}=3x^2+28(2x+1)$$
So, I get $152$. What should I do? Thank you!

The answer is that it depends (slightly) on the intent of the problem.  When I read the problem as you've written it, I see this:

Given that $y = 2x+1$ and  $f(x,y) = x^{3} + 7y^{2},$ calculate $\frac{\partial f(2,3)}{\partial x}$.

In that statement, I interpret this as $f$ is a function of two variables, $x$ and $y$, and we are asked to compute $\frac{\partial f}{\partial x}(x,y)$ and evaluate the result at $(x,y) = (2,3)$.  In this case, the fact that $y$ depends on $x$ isn't relevant, because since $f$ depends explicitly on $x$ and $y$, $\frac{\partial f}{\partial x}(x,y)$ mean hold $y$ constant and differentiate with respect to $x$.  That gives us $$\frac{\partial f}{\partial x}(x,y) = 3x^{2}\implies \frac{\partial f(2,3)}{\partial x} = 12.$$
Now, if instead the problem was phrased as

Given that $y = 2x+1$ and  $f(x,y) = x^{3} + 7y^{2},$ calculate the total rate of change of $f$ with respect to $x$ at $(2,3)$.

In this case, I would interpret the problem as asking for the total derivative, which accounts for dependencies among the variables.  In this case, your answer would be $$\frac{df}{dx} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y}\frac{dy}{dx} = 3x^2+28(2x+1) \implies \frac{df}{dx}(2,3) = 152.$$
If the original problem does indeed ask for $\frac{\partial f(2,3)}{\partial x}$ then I would say the first interpretation is correct.  If you are adding that notation yourself, it may be the case that the second interpretation is correct.
