Using the chain rule in $\mathbb{R}^n$

Asked Apr 13 '14 at 17:23

Active Apr 13 '14 at 17:28

Viewed 33 times

Suppose that the function $\psi:\mathbb{R}^2 \to \mathbb{R}$ is continuously differentiable. Define the function $g:\mathbb{R}^2 \to \mathbb{R}$ by $g(s,t) = \psi(s^2t,s)$ for $(s,t) \in \mathbb{R}^2$. Find $\frac{\partial g}{\partial s}(s,t)$.

Here is what I have writtens as the solution but I am not positive I am applying the chain rule correctly:

$\frac{\partial g}{\partial s}(s,t) = \frac{\partial \psi}{\partial s}(s^2t, t)2st+\frac{\partial \psi}{\partial t}(s^2t, t)$.

edited Apr 13 '14 at 17:28

asked Apr 13 '14 at 17:23

Zachary Luety

You might explicitly point out that $\frac{\partial \psi}{\partial s}$ is evaluated at $(s^2t,s)$. – Giuseppe Negro Apr 13 '14 at 17:26
Made the correction. Thanks – Zachary Luety Apr 13 '14 at 17:29
2

I would tend to opt for $D_{1} \phi$ and $D_{2} \phi$ to reflect that it is the change with respect to the first and second variables. – Christopher K Apr 13 '14 at 17:32
Like this? $D_1\psi(2st)+D_2\psi$ – Zachary Luety Apr 13 '14 at 17:40

Using the chain rule in $\mathbb{R}^n$

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