A basic question about partial differentiation

F is a smooth function of x and y, i.e. F(x,y). If $H(x)=F(x,0)$, when can I have $\dfrac{\partial F}{\partial x}(x,0)= \dfrac{dH} {dx}(x)?$

I think we can show this equality from the definition of partial differentiation easily. Is this equality true for all functions of (x,y) which are first order differentiable with respect to x?

Thanks

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If you set $H(x)=F(x,c)$, for any $c\in\mathbb{R}$, then $$\frac{dH}{dx}(x)=\lim_{t\to0}\frac{H(x+t)-H(x)}{t}=\lim_{t\to0}\frac{F(x+t,c)-F(x)}{t}=\frac{\partial F}{\partial x}(x,c)$$ so the existence of the derivative of $H$ in the point $x$ is equivalent to the existence of the partial derivative of $F$ in the direction $x$ in the point $(x,c)$.