# Multivariable Chain-Rule Problem

## Homework Statement

Let g(x, y) = f(sin(y), cos(x)). Find the second partial derivative of g with respect to x (g_xx).

## The Attempt at a Solution

I attempted to find g_x, but I'm not entirely sure how chain rule applies in this situation.

Is this correct?

g_x = f_x(sin(y), cos(x)) * (-sin(x))

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To get the derivative with respect to $x,$ by chain rule: $$\frac{\partial g}{\partial x}=\frac{\partial f}{\partial x}\frac{\partial \sin y}{\partial x}+\frac{\partial f}{\partial y}\frac{\partial \cos x}{\partial x}$$
and see every term if one can be cancelled.

To get the derivative with respect to $x,$ by chain rule: $$\frac{\partial g}{\partial x}=\frac{\partial f}{\partial x}\frac{\partial \sin y}{\partial x}+\frac{\partial f}{\partial y}\frac{\partial \cos x}{\partial x}$$
and see every term if one can be cancelled.
Ah, thank-you. The first term cancels, right?

Ah, thank-you. The first term cancels, right?
Yes, for the $\sin y$ doesn't depend on $x.$

However, forgetting to remind, in the calculation, mind the terms regarding $f$ you have to plug in its pair $(\sin y,\cos x).$

However, forgetting to remind, in the calculation, mind the terms regarding $f$ you have to plug in its pair $(\sin y,\cos x).$
Should it be d/dx(siny) as in the single var. case?

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To be precise, you are right, but using partial derivative doesn't make no sense based on the definition also haha (just for my laziness)

To be precise, you are right, but using partial derivative doesn't make no sense based on the definition also haha (just for my laziness)
How would I obtain g_y though? I'm also not sure how to derive the first formula.

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Well...then maybe you don't understand the chain rule totally...For reference: https://en.wikipedia.org/wiki/Chain_rule
To make yourself really acquire the information, I suggest getting the full picture of it including the proof.

Well...then maybe you don't understand the chain rule totally...For reference: https://en.wikipedia.org/wiki/Chain_rule
To make yourself really acquire the information, I suggest getting the full picture of it including the proof.
The way I learned the chain rule (in the context of multivariable functions) was to draw a dependency diagram. In this case, however, the dependency diagram is not clear.

Ray Vickson
Homework Helper
Dearly Missed

## Homework Statement

Let g(x, y) = f(sin(y), cos(x)). Find the second partial derivative of g with respect to x (g_xx).

## The Attempt at a Solution

I attempted to find g_x, but I'm not entirely sure how chain rule applies in this situation.

Is this correct?

g_x = f_x(sin(y), cos(x)) * (-sin(x))
When I learned this stuff we were advised to use notation like $f_1$, which is "partial derivative of $f$ with respect to the first variable" and $f_2$ for "... with respect to the second variable". That way, when you swap the positions of $x$ and $y$ (as done in this problem) you avoid getting yourself hopelessly confused.