# How Do You Tackle Partial Differentiation with Trigonometric Functions?

• phrygian
In summary, for the given problem of finding dx/dy and d^2x/dy^2 when x = yz and y = 2sin(y+z), the approach is to first find dz in terms of dy from the second equation and then plug it into the first equation. From there, solve for dx and divide both sides by dy to find the desired derivatives. The use of inverse trig functions is not necessary in this problem.
phrygian

## Homework Statement

If x = yz and y = 2sin(y+z), find dx/dy and d^2x/dy^2

## The Attempt at a Solution

I am beyond confused at how to even start this one, the problem is not like any of the examples in my book.

I know x = ydz + zdy but don't know how to deal with z and dz?

Thanks for the help

Find dz in terms of dy from the second equation and plug it into the first: dx=ydz+zdy.

Is it correct to use inverse trig function for that?

I can see no reason to. The second equation is y= 2sin(y+z) so dy= 2cos(y+z)(dy+ dz). From dx= zdy+ ydz, (NOT the "x= zdy+ ydz" you give. I assume that was a typo.), we get ydz= dx- zdy so dz= (1/y)dx- (z/y)dy. Putting that into the second equation, dy= 2cos(y+z)(dy+ (1/y)dx- (z/y)dy)= 2cos(y+z)(1- (z/y)dy+ (2/y)cos(y+z)dx. Solve that for dx and divide both sides by dy.

You can do anything you want that doesn't violate the laws of math!

## 1. What is partial differentiation and how is it different from ordinary differentiation?

Partial differentiation is a mathematical concept used to find the rate of change of a function with respect to one of its variables, while holding all other variables constant. It is different from ordinary differentiation in that ordinary differentiation is used to find the rate of change of a function with respect to a single variable.

## 2. How is partial differentiation used in science?

Partial differentiation is used in science to analyze and understand systems with multiple variables, such as in physics, economics, and engineering. It allows scientists to study how a certain variable affects the outcome of a system while keeping all other variables constant.

## 3. Why is partial differentiation confusing?

Partial differentiation can be confusing because it involves multiple variables and can require understanding of advanced mathematical concepts, such as limits and partial derivatives. It also requires a clear understanding of the problem at hand and the variables involved.

## 4. What are some common mistakes made when using partial differentiation?

Some common mistakes made when using partial differentiation include forgetting to hold all other variables constant, using the wrong notation for partial derivatives, and not properly simplifying the final result. It is important to pay close attention to the problem and double check all steps to avoid these mistakes.

## 5. How can I improve my understanding of partial differentiation?

To improve your understanding of partial differentiation, it is important to have a strong foundation in calculus and to practice solving various problems. You can also seek help from a tutor or teacher, as well as consult online resources and textbooks for additional explanations and examples.

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