Assume we have two functions ($x$ and $y$ are independent):
$$f(x,y)$$ and $$f_{x}(y)$$
The second function derives from the first one, it just fixes $x$ and only depend on y.
In this case, do we have the following equation:
$$\frac{\partial f(x,y)}{\partial y} = \frac{df_{x}(y)}{dy}$$
Convert a partial derivative to an ordinary derivative?
For a fixed $x$ you define $f_x(y) = f(x,y)$, so in fact the definition of the partial derivative is $$ \frac{\partial f}{\partial y}(x,y) = \lim_{h \to 0} \frac{f(x,y+h) -f(x,y)}{h} = \lim_{h \to 0} \frac{f_x(y+h) - f_x(y)}{h} = \frac{d f_x}{dy}(y). $$ So, yes, the partial derivative of $f$ in the $y$ direction is the total derivative of $f_x$.