Chain rule to calculate partial derivative of a given function

Question

The question I am working on is asking- Use the chain rule to calculate $\frac{\partial F}{\partial s}$ and $\frac{\partial F}{\partial t}$ for the given function F:

$$F(s,t)=f(x(s,t),y(s,t))$$ where $$f(x,y)=2x^2y-xy^2$$ $$x(s,t)=t^2+3s^2$$ and $$y(s,t)=3st$$

This is what I have done

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$\frac{\partial F}{\partial s}$=$\frac{\partial F}{\partial x}$*$\frac{\partial x}{\partial s} = (4xy-y^2)*(6s)$

and

$\frac{\partial F}{\partial t}$=$\frac{\partial F}{\partial y}$*$\frac{\partial y}{\partial t} = (2x^2-2yx)*(3s)$

Is my work correct ?

Answer 1

Hint: When dealing with partial fractions, all terms must come into account. For example: $$\frac{\partial F}{\partial s}(s,t) = \frac{\partial f}{\partial x}(x(s,t),y(s,t))\cdot \frac{\partial x}{\partial s}(s,t)+\frac{\partial f}{\partial y}(x(s,t),y(s,t))\cdot \frac{\partial y}{\partial s}(s,t).$$ Now just plug in everything there. For $\frac{\partial F}{\partial t}(s,t)$ is is the same thing.

Answer 2

This question isn't too bad, but when I was learning partial derivatives, our teacher would craft the most indiscernible families of functions. I found this diagram useful as I was learning:

How to draw it:

1. Start with the top node (e.g. $F$), a variable that is not a dependent in other functions.
2. Then, draw nodes below it for each variable it depends on (e.g. $F(x,y)$), and connect them.
3. Once you have a node for every variable, write the partial derivative along each line (N.B. if a node only has one subnode, you should use $d$ not $\partial$)

Suppose we wanted to find $\frac{\partial F}{\partial s}$

1. Trace each route on the graph from $F$ to $s$.
2. Multiply the terms on the lines along each route.
3. Add the products from each route.

So, we have: $$\frac{\partial F}{\partial s}=\frac{\partial F}{\partial x}\frac{\partial x}{\partial s} + \frac{\partial F}{\partial y}\frac{\partial y}{\partial s}$$ If you use this diagram correctly and frequently, you won't miss any dependencies and you'll soon hate drawing it so much you'll learn to write differentials without it.