What is the partial derivative of u with respect to t in terms of x, y, and t?

In summary, the conversation discusses the use of partial derivatives to calculate the quantity u = e^{-xy} for a given path (x,y) in time t. The participant is unsure of how to proceed after partially differentiating u and asks for guidance. They are then reminded to use the chain rule to calculate \frac{du}{dt} as a function of x, y, and t.
  • #1
jamesbob
63
0
This is annoying me as i have the answer on the tip of my pen, just can't write it down. I'm not 100% sure i understand what the question is asking me to do.

Consider the quantity [tex] u = e^{-xy} [/tex] where (x,y) moves in time t along a path:

[tex]x = \cosh{t}, \mbox{ } y = \sinh{t}[/tex]​

Use a method based on partial derivatives to calculate [tex] \frac{du}{dt} [/tex] as a function of x, y and t.

My answer:

I partially differentiated u, getting:

[tex] \frac{\delta{u}}{\delta{x}} = -ye^{-xy} [/tex]
[tex] \frac{\delta{u}}{\delta{y}} = -xe^{-xy} [/tex]
So does this mean [tex] du = -ye^{-xy} + -xe^{-xy} ? [/tex]

I though that i would get du from the part iv just explained, then get dt from differentiating x and y. But this ofcourse leaves me with expressions for dx/dt and dy/dt. Where do i go from here?
 
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  • #2
Use the chain rule:
[tex]\frac{du}{dt} = \frac{\partial u}{\partial x} \frac{dx}{dt} + \frac{\partial u}{\partial y} \frac{dy}{dt}[/tex]
 
Last edited:
  • #3
Yeah, realized that after some research - just something id never saw. Dead easy tho. Thanks anyway :smile:
 

1. What is a partial derivative?

A partial derivative is a mathematical concept used to measure the rate of change of a function with respect to one of its variables while holding all other variables constant. It is represented by the symbol ∂ and is commonly used in multivariable calculus.

2. How is a partial derivative calculated?

To calculate a partial derivative, you take the derivative of the function with respect to the variable of interest while treating all other variables as constants. This can be done by using the standard rules of differentiation, such as the power rule or product rule.

3. What is the purpose of using partial derivatives?

Partial derivatives are useful for understanding how a function changes in relation to specific variables, and can be used to find maximum or minimum values of a function. They are also used in fields such as physics and economics to model complex systems.

4. 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, while holding all other variables constant. A total derivative, on the other hand, takes into account the changes in all variables that affect the function. In other words, a total derivative takes into account both the partial derivatives and the cross-derivatives.

5. Can you give an example of a real-life application of partial derivatives?

One example of a real-life application of partial derivatives is in economics, where they are used to model the relationship between multiple variables, such as supply and demand. Partial derivatives can be used to find the optimal price for a product by determining the rate of change of demand with respect to price, while holding other factors constant.

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