Implicit functions can be tricky to deal with when we use traditional ways of finding derivatives. This is because they don’t look like the usual mathematical expressions that we are used to.

When we have explicit functions, like $y = f(x)$, we can easily find derivatives. We can apply rules like the power rule, product rule, or quotient rule. But implicit functions, shown as $F(x, y) = 0$, require us to think about the relationship between the variables in a different way.

To find the derivative $dy/dx$ for an implicit function, we use something called implicit differentiation. This method involves taking the derivative of both sides of the equation with respect to $x$. We treat $y$ as if it depends on $x$.

For example, when we differentiate $F(x, y)$, we get:

$\frac{dF}{dx} = \frac{\partial F}{\partial x} + \frac{\partial F}{\partial y} \frac{dy}{dx} = 0$

From this equation, we can figure out $\frac{dy}{dx}$.

But this isn't as simple as it might seem. It not only includes $\frac{\partial F}{\partial x}$, but also $\frac{\partial F}{\partial y}$. This makes the calculation a bit more complicated.

This shows us a key challenge: when we work with implicit functions, we must adapt our techniques. We have to consider how $y$ changes as $x$ changes.

In short, implicit functions need us to look at problems from a different angle and use a more complicated way of finding derivatives. This shows just how flexible and deep calculus can be.