I am bit confused regarding the geometrical/logical meaning of partial and total derivative. I have given my confusion with examples as follows
Question
Suppose we have a function $f(x,y)$ , then how do we write the limit method of representing $ \frac{\partial f(x,y)}{\partial x} \text{and} \frac{\mathrm{d} f(x,y)}{\mathrm{d} x}$ at (a,b)? What is the difference?
Imagine I have a function $f(t,x,y)= t^4+x^2+y^3+2xy^9\tag 1$
what could be $ \frac{\partial f(t,x,y)}{\partial x} , \frac{\mathrm{d} f(t,x,y)}{\mathrm{d} x},\frac{\partial f(t,x,y)}{\partial t} , \frac{\mathrm{d} f(t,x,y)}{\mathrm{d} t}$? What is the difference between them in meaning?
Imagine I have a function $f(t,x,y)= t^4+x^2+y^3+2xy^9,x=\psi(t),y=\tau(t)\tag 1$
what could be $ \frac{\partial f(t,x,y)}{\partial x} , \frac{\mathrm{d} f(t,x,y)}{\mathrm{d} x},\frac{\partial f(t,x,y)}{\partial t} , \frac{\mathrm{d} f(t,x,y)}{\mathrm{d} t}$? What is the difference between them in meaning?
When we can say $ \frac{\partial f(x,y)}{\partial x} = \frac{\mathrm{d} f(x,y)}{\mathrm{d} x}$? Can we say $\frac{\partial^2 f(t,x,y)}{\partial x dy} =\frac{\partial^2 f(t,x,y)}{ dy \partial x}$ if function $f(t,x,y)$ is $C^2$ continuous?
We have a function curve ${f(t,g_1,g_2,g_3)}_{3 \times 1} \tag 2$
and we have a $4 \times 1$ array called $p=\begin{pmatrix}t\\ g_1\\ g_2\\ g_3 \end{pmatrix}$ All t and $g_i$ are variables, t is the curvilinear parameter . What is the meaning difference between $\frac{\partial f(t,g_1,g_2,g_3)}{\partial p} \text{and} \frac{d f(t,g_1,g_2,g_3)}{d p} $? How do we express the difference in limit notations?
NB : Replies written with question enumeration will be more helpul. These are basics, I know. But I get confused some time. Looking for interpretations beyond equations. Means geometrically or logically coherent one.This is not a homework problem just because it is simple. I made the example for expressing my issue so that I can learn from the result. Thanks for the time to read my question.