Using Chain Rule to Solve Questions - A Step-by-Step Guide

[EconometricAli]

Hey everyone, could anyone help me figure out how to use chain rule to solve these questions in the attachments below?

 

[HOI]

Okay your second picture gives the "chain rule":
If f is a function of x, y, and z and x, y, and z are functions of the variables, s and t, then f is a function of s and t and $\frac{\partial f}{\partial t}= f_x\frac{\partial x}{\partial t}+ f_y\frac{\partial y}{\partial t}+ f_z\frac{\partial z}{\partial t}$ and $\frac{\partial f}{\partial s}= f_x\frac{\partial x}{\partial s}+ f_y\frac{\partial y}{\partial s}+ f_z\frac{\partial z}{\partial s}$.
Great!

Now, in the first problem, $f(x, y, z)= xyz^{10}$, $x= t^3$, $y= ln(s^2\sqrt{t})$, and $z= e^{st}$.

The first thing I would do is write $y= 2(1/2)ln(st)= ln(st)$.

Now $f_x= yz^{10}$ and $\frac{\partial x}{\partial t}= 3t^2$ so $f_x\frac{\partial x}{\partial t}= 3t^2yz^{10}$. If you want to reduce that to s and t only, replace y with $ln(st)$ and z with $e^{st}$ to get $f_x\frac{\partial x}{\partial t}= 3t^3ln(sy)e^{10st}$.

$f_y= xz^{10}$ and $\frac{\partial y}{\partial t}= \frac{s}{t}$1 so $f_y\frac{\partial y}{\partial t}= xz^{10}\frac{s}{t}=\frac{st^3e^{10st}}{t}= st^2e^{10t}$.

$f_z= 10xyz^9$ and $\frac{\partial z}{\partial t}=se^{st}$ so $f_z\frac{\partial z}{\partial x}= 10xyz^9(se^{st})= 10(t^3)ln(st)e^{9st}(se^{st})= 90st^3ln(st)e^{10st}$.

$\frac{\partial f}{\partial t}$ is the sum of those.

$\frac{\partial f}{\partial s}$ is done the same way but with $\frac{\partial x}{\partial s}$, $\frac{\partial y}{\partial s}$, and $\frac{\partial z}{\partial s}$.

 

[skeeter]

The first thing I would do is write $y=2(1/2)ln(st)=ln(st)$

$y = \ln(s^2 \cdot \sqrt{t}) = 2\ln(s) + \dfrac{1}{2}\ln(t)$

 

[HOI]

One of these days I really need to learn Math!

 

[EconometricAli]

Thank you for the response but I'm lost in part b. The portion where fx (dx/dt) for x^10+y^11+z^10... what is its final form?

 

[HOI]

Okay, "part b" has $f= x^{10}+ y^{11}+ z$, $x= \sqrt{t+ s^2}$, $y=\frac{t}{s}$, $z= st$ and you want to find $\frac{\partial f}{\partial s}$ and $\frac{\partial f}{\partial t}$.

What are
$\frac{\partial f}{\partial x}$
$\frac{\partial f}{\partial y} $
$\frac{\partial f}{\partial z}$ ?

What are
$\frac{\partial x}{\partial s}$
$\frac{\partial x}{\partial t}$

$\frac{\partial y}{\partial s}$
$\frac{\partial y}{\partial t}$

$\frac{\partial z}{\partial s}$
$\frac{\partial z}{\partial t}$ ?

If you know basic Calculus you should be able to answer those.
If you can't do some of them, tell us which.

The "chain rule" says
$\frac{\partial f}{\partial s}= \frac{\partial f}{\partial x}\frac{\partial x}{\partial s}+ \frac{\partial f}{\partial y}\frac{\partial y}{\partial s}+ \frac{\partial f}{\partial z}\frac{\partial z}{\partial s}$
and
$\frac{\partial f}{\partial t}= \frac{\partial f}{\partial x}\frac{\partial x}{\partial t}+ \frac{\partial f}{\partial y}\frac{\partial y}{\partial t}+ \frac{\partial f}{\partial z}\frac{\partial z}{\partial t}$

 

[EconometricAli]

I got my answers marked and lost a lot of marks actually. I'm not sure why.