Let $f: \mathbb{R^2} \to \mathbb{R^2}$ be a differentiable function in $(0,0)$ and $\gamma_1, \gamma_2 :[a,b] \to \mathbb{R^2}$ differentiable paths such that $\gamma_1(0)=\gamma_2(0)=(0,0)$ and we know $\gamma_2 (t)=\gamma _1(t)+(t^2,t^3) $
I need to show that $(i)$ $\frac{d(f(\gamma_1 (t)))}{dt}|_{t=0}=\frac{d(f(\gamma_2 (t)))}{dt}|_{t=0}.$
I defined $g1=f((\gamma_1(t))) $ and $ g2=f((\gamma_2(t)))$, If I'm not mistaken I need to show that $\frac{d g_1}{dt}(0)=\frac{dg_2}{dt}(0)$, and If I'm not mistaken $(ii)$ $\frac{d g_1}{dt}=\frac{df}{dx}(\gamma_1(0))\frac{dx_1}{dt}+\frac{df}{dy}(\gamma_1(0))\frac{dy_1}{dt},$ $\frac{d g_2}{dt}=\frac{df}{dx}(\gamma_2(0))\frac{dx_2}{dt}+\frac{df}{dy}(\gamma_2(0))\frac{dy_2}{dt},$ (Are these right definitons?)
where $\gamma_1(t)=(x_1(t),y_1(t))$, $\gamma_2(t)=(x_1(t),y_2(t))$ (Did I get right these as well?)
Now we know as well $\frac{dx_2 }{dt}=\frac{dx_1}{dt}+t, \frac{dy_2 }{dt}=\frac{dy_1}{dt}+3t^2,$
Can I just plug in $\frac{dx_2 }{dt}(0)=\frac{dx_1}{dt}(0)+0, \frac{dy_2 }{dt}(0)=\frac{dy_1}{dt}(0)+30^2,$ and then use it in and conclude that $(ii)$ is correct and due to that $(i)$ is correct also?
What do I do with all this information? I think I got it somehow wrong. I'd really love your guidance.
Thanks a lot!