Problem
Let $f\left ( x,y \right )$ be a differentiable function on $\mathbb{R}^{2}$.
Find a formula for $\frac{d}{dt}f\left ( t,t^{2} \right )\mid _{t=1}$ in terms of the partial derivatives: $\frac{\partial f}{\partial x}\left ( 1,1 \right )$ and $\frac{\partial f}{\partial y}\left ( 1,1 \right )$
I started to solve the problem as follows: I applied the chain rule, and then I got: $\frac{d}{dt}f\left ( t,t^{2} \right )=f_{1}\left ( t,t^{2} \right )\cdot 1+f_{2}\left ( t,t^{2} \right )\cdot 2t$ and as $t\rightarrow 1$, I get: $\frac{d}{dt}f\left ( t,t^{2} \right )\Big| _{t=1}=\frac{\partial f}{\partial x}\left ( 1,1 \right )+2\frac{\partial f}{\partial y}\left ( 1,1 \right ).$
Is my answer correct?