- #1
MatinSAR
- 606
- 183
- Homework Statement
- Find partial derivatives
- Relevant Equations
- dy/dx=(dy/dt)(dt/dx)
Can someone please help me to find out what happened here ?
Not understanding these manipulations involving Partial Derivatives
- Thread starter MatinSAR
- Start date
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.
- #2
ergospherical
- 1,055
- 1,347
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
$$\partial f(u,v) / \partial t = (\partial f/ \partial u) (\partial u/ \partial t) + (\partial f/ \partial v) (\partial v/ \partial t)$$
$$\partial f(u,v) / \partial t = (\partial f/ \partial u) (\partial u/ \partial t) + (\partial f/ \partial v) (\partial v/ \partial t)$$
- #3
MatinSAR
- 606
- 183
That "tx" confused me ...ergospherical said:
Now it's clear...
Thank you for your time
FAQ: Not understanding these manipulations involving Partial Derivatives
What is a partial derivative?
A partial derivative is a derivative taken of a multivariable function with respect to one variable while keeping the other variables constant. It measures how the function changes as one particular variable changes, providing insight into the function's behavior in a specific direction.
How do you compute a partial derivative?
To compute a partial derivative, treat all other variables as constants and differentiate the function with respect to the variable of interest. For example, if you have a function f(x, y), the partial derivative with respect to x is found by differentiating f with respect to x while treating y as a constant.
Why do we need partial derivatives?
Partial derivatives are essential in multivariable calculus because they help us understand how a function changes in each individual direction. They are used in optimization problems, in finding tangent planes to surfaces, and in various applications in physics, engineering, and economics where functions depend on multiple variables.
What is the difference between a partial derivative and a total derivative?
A partial derivative measures the rate of change of a function with respect to one variable, keeping other variables constant. In contrast, a total derivative takes into account the rates of change with respect to all variables and their interdependencies. The total derivative provides a comprehensive rate of change considering all variables simultaneously.
Can you explain the concept of higher-order partial derivatives?
Higher-order partial derivatives are derivatives taken multiple times with respect to one or more variables. For instance, the second-order partial derivative with respect to x is found by differentiating the first-order partial derivative with respect to x again. Mixed partial derivatives involve differentiating with respect to different variables in succession, like taking the derivative with respect to x and then with respect to y.