# How can partial derivatives feature in the definition of a function?

I have a map $f(t,g,h)$ where $f:[0,1]\times C^1 \times C^1 \to \mathbb{R}.$

I want to define $$F(t,g,h) = \frac{d}{dt}f(t,g,h)$$ where $g$ and $h$ have no $t$-dependence in them. So $g(x) = t^2x$ would not be admissible if you want to calculate what $F$ is. How do I write this properly? Is it correct to write instead

Define $F$ by $F(t,\cdot,\cdot) = \frac{d}{dt}f(t,\cdot,\cdot)$.

But there is some ambiguity in the arguments. What is the best way to write it?

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$$F(t,g,h) = \frac{\partial}{\partial t} f(t,g,h)$$
The definition of a partial derivative is that you differentiate with respect to one variable as usual while keeping all others fixed. The different kind of $d$ is there to remind you of that, because in this case writing $\frac{d}{dt}f(t,g,h)$ would imply that $g$ and $h$ have some kind of $t$-dependence.
Since $g$ and $h$ do not depend on $t$ one has $$F(t,g,h) := \frac{d}{dt}f(t,g,h)=f_{.1}(t,g,h)\ ,$$ where $f_{.1}$ denotes differentiation with respect to the first variable, whatever its name.