Partial Differentiation help

In summary, the conversation is about someone asking for help with finding the first partial derivatives of a function involving exponentials and variables u and v. The provided answer uses a specific chain rule and is confirmed to be correct.
  • #1
elle
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Partial Differentiation help please!

Hi, I was wondering if anyone is doing degree level maths who can help me with the following question. Thanks very much!

I was asked to find the first partial derivatives of z (in terms of x and y) with respect to x and y where:

z = e^(uv) where u = x -y and v = xy

The answer I got was:

∂w/∂x = (2xy-y²) e^[(x-y)(xy)]

and ∂w/∂y = (x² - 2xy) e^[(x-y)(xy)]

Can anyone help me confirm if that is right? I've used a particular chain rule i was given in my notes to approach the answer. Thanks again
 
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  • #2
They look good to me!
 

FAQ: Partial Differentiation help

What is partial differentiation?

Partial differentiation is a mathematical process used to calculate the rate of change of a function with respect to one of its variables, while holding all other variables constant. It is used in multivariable calculus and is an important tool in analyzing the behavior of functions with more than one independent variable.

Why is partial differentiation important?

Partial differentiation allows us to study how a function changes when only one variable is changed, while keeping all other variables fixed. This is useful in many fields, such as economics, physics, and engineering, where functions often have multiple independent variables that affect the outcome.

How is partial differentiation different from ordinary differentiation?

Ordinary differentiation, also known as single-variable differentiation, calculates the rate of change of a function with respect to one variable. Partial differentiation, on the other hand, calculates the rate of change of a function with respect to one variable while holding all other variables constant.

What are some common applications of partial differentiation?

Partial differentiation is used in a variety of fields, such as optimization problems in economics, finding rates of change in physics, and determining critical points in engineering. It is also commonly used in analyzing surfaces and curves in three-dimensional space.

What do the partial derivative symbols (∂) and (∂y/∂x) represent?

The symbol (∂) represents a partial derivative, while (∂y/∂x) represents the partial derivative of y with respect to x. This notation indicates that we are only interested in the rate of change of the function with respect to one variable, while treating all other variables as constants.

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