Question: 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?

Answer: 
Without words... Well, nearly without words...

![enter image description here](https://i.stack.imgur.com/la68x.png)
For $0\leq\phi\leq{\pi\over2}$ consider the two points $(\cos \phi,\pm\sin \phi)$ on the unit circle. Their distance is $2\sin \phi$, while the length of the circular arc between them is $2\phi$. Therefore we have
$$2\sin\phi\leq 2\phi\qquad(0\leq\phi\leq{\pi\over2})\ .$$
Putting $\phi:=|x|$ we obtain
$$|\sin x|\leq |x|\qquad(0\leq|x|\leq{\pi\over2})\ ,$$
and for $|x|\geq{\pi\over2}$ one obviously has $|x|\geq{\pi\over2}>1\geq|\sin x|$.