I am looking to apply the Implicit Function Theorem to a problem I am working on, and I am unsure if I'm considering the partial derivative correctly. I have a function, say $ h(x,y) = b - x^2 + y$, in which there is an explicit functional relationship $b(x)= 2x^3$. I need to know the partial derivative of $h$ with respect to $x$. My thinking is as follows:
- If I plugged in $b(x)$, then I'd have $\frac{\partial h}{\partial x} = 6x^2 - 2x $.
- But, I know in a partial derivative, I'm supposed to ignore dependencies between variables, so I should ignore $\frac{\partial b}{\partial x}$.
- If I ignore $\frac{\partial b}{\partial x}$, then $\frac{\partial h}{\partial x} = 2x$.
- But I shouldn't get a different answer of how $h$ depends on $x$, holding $y$ constant, based on whether I call the first term $b$ or $2x^3$.
Is it that partial derivatives only ignore implicit relationships? Or is it because $b$ is not an input to the function?
EDIT: Going to the limit definition of the partial derivative, Does it really just depend on the domain (in the sense of what variables are counted as inputs) of the function? If $b$ is not a separate input to the function, then any change in $b$ due to a change in $x$ is just part of the overall effect of changing $x$. If we were going to allow for $b$ potentially taking on some values other than those given by $b(x)$, then it would make sense to include $b$ an input to the function, but since we only care about $b=b(x)$, that would not make sense.