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I am having a difficult time to understand the chain rule, and I have this exercise:

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ to be a differentiable function. Define $\psi(x,y)=f(xy,x^2y^2)$. How to compute the partial derivatives and how to proof that the directional derivative $D_{(-x,y)}\psi(x,y)=0$?

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$$\psi_x=y\cdot f_x+2xy^2\cdot f_y$$ $$\psi_y=x\cdot f_x+2x^2y\cdot f_x$$

and we have

$$D_{(-x,y)}\psi=-x\cdot \psi_x+y\cdot\psi_y=0$$

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