Not understanding these manipulations involving Partial Derivatives

In summary, The conversation discusses differentiating a function ##f## with respect to its arguments and then differentiating the arguments with respect to ##t##. To make it clearer, it is suggested to write ##u = tx## and ##v = ty##, and then the formula for the partial derivative with respect to ##t## is given. The person in the conversation initially found the notation confusing, but it is now clear.
  • #1
MatinSAR
520
174
Homework Statement
Find partial derivatives
Relevant Equations
dy/dx=(dy/dt)(dt/dx)
Can someone please help me to find out what happened here ?

1675445974557.png
 
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  • #2
It's differentiating ##f## with respect to its arguments, then differentiating the arguments with respect to ##t##. It might be clearer if you write ##u = tx## and ##v=ty##, then
$$\partial f(u,v) / \partial t = (\partial f/ \partial u) (\partial u/ \partial t) + (\partial f/ \partial v) (\partial v/ \partial t)$$
 
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  • #3
ergospherical said:
It's differentiating ##f## with respect to its arguments, then differentiating the arguments with respect to ##t##. It might be clearer if you write ##u = tx## and ##v=ty##, then
$$\partial f(u,v) / \partial t = (\partial f/ \partial u) (\partial u/ \partial t) + (\partial f/ \partial v) (\partial v/ \partial t)$$
That "tx" confused me ...
Now it's clear...
Thank you for your time 🙏🙏
 
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1. What are partial derivatives and why are they important in science?

Partial derivatives are a mathematical tool used to calculate the rate of change of a function with respect to one of its variables, while holding all other variables constant. They are important in science because they allow us to analyze how changes in one variable affect the overall behavior of a system or process.

2. How do you calculate a partial derivative?

To calculate a partial derivative, you first need to identify the variable you want to differentiate with respect to. Then, you treat all other variables as constants and use the standard rules of differentiation to find the derivative of the function with respect to the chosen variable.

3. What is the difference between partial derivatives and ordinary derivatives?

The main difference between partial derivatives and ordinary derivatives is that partial derivatives consider changes in only one variable, while ordinary derivatives consider changes in all variables. In other words, partial derivatives measure the slope of a function in one direction, while ordinary derivatives measure the slope in all directions.

4. How are partial derivatives used in real-world applications?

Partial derivatives are used in a wide range of real-world applications, including physics, engineering, economics, and statistics. They can be used to analyze the behavior of complex systems, optimize processes, and make predictions about future outcomes.

5. What are some common mistakes when working with partial derivatives?

Some common mistakes when working with partial derivatives include forgetting to treat other variables as constants, using the wrong rules of differentiation, and not simplifying the final expression. It is important to carefully follow the steps and double-check your work to avoid these common errors.

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