How to Solve a Derivative Presented in a Non-Standard Format?

[Maniac_XOX]

Homework Statement

I've submitted a similar thread previously but messed it all up so here's a second try. The objective is to find the first and second partial derivative of the function z = tan x^2*y^2

Relevant Equations

tan x^2*y^2 = (sin x^2*y^2)/(cos x^2*y^2)
quotient rule: if z= f/g, z' = (f'*g - f*g')/g^2

I have attached a word document demonstrating the working out cos i was too lazy to learn how Latex primer works and writing it like I did above would've been too hard too read. I tried to make it as understandable as possible, presenting fractions as
' a ' instead of ' a / b ' .
------
b

 

[Mark44]

I was able to open the Word document, but your work starts off with an error.
From the Word doc:

Since I started with finding the derivative in respect to x, throughout the calculation I treated y^2 as constant 'a'

No, that's not correct. Your function is $z = \tan(x^2y^2)$, so z is a function of both x and y. Unless both x and y are themselves functions of some other variable (such as t, for example), there is no derivative dz/dx.

In this case there are two first (partial) derivatives: $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$. And there are four second partials $\frac{\partial^2 z}{\partial x^2}$, $\frac{\partial^2 z}{\partial y^2}$, and the two mixed partials. You were told all of this in the other thread you started.

The two first partials aren't too hard if you leave $\tan(x^2y^2)$ as it is and don't write it as a quotient. You do know the formula for the derivative of tan(x), don't you? It greatly complicates matters to take the derivative of sin(x)/cos(x).

One more thing. Please use parentheses around the arguments to any trig function. I realize that textbooks often write things like $\sin \theta$, but as soon as the function arguments get more complicated, it really helps to add in the parentheses. tan x^2*y^2 is ambiguous, but tan(x^2*y^2) is perfectly clear.

 

[Maniac_XOX]

Is your function $\tan(x^2) + y^2 = y^2 + \tan x^2?$ That is what you wrote. If you mean $\tan(x^2+y^2)$ you need to write parentheses, to keep the things sorted out properly.

You do not need to use LaTeX (although it will be worth your effort in the long run); you can just use plain typwriter text, like this: tan(x^2 +y^2). That is perfectly clear to everybody. You can even write things like (d/dx) tan(x^2+y^2), to indicate a partial derivative wrt x.

I was not able to get the word file to open properly on my laptop, and not open at all on my i-phone. You should avoid posting word attachments if you are really serious about wanting help

No the * sign stands for multiplication so it's $\tan (x^2 * y^2)$

 

[Mark44]

Thread closed as duplicate.
@Maniac_XOX, please continue with the other thread you started - https://www.physicsforums.com/threads/calculus-with-two-variables.969336/