Chain rule confusion partial derivatives

In summary: But, remember, w does not have a t variable, so \frac{\partial w}{\partial t} is 0. Therefore:\frac{dw}{dt} = \frac{\partial w}{\partial x} \frac{dx}{dt} + \frac{\partial w}{\partial y} \frac{dy}{dt} + 0Since w = xy + yz^2, we can find the partial derivatives:\frac{\partial w}{\partial x} = y\frac{\partial w}{\partial y} = x + z^2Plugging these in, we get:\frac{dw}{dt} = y \frac{dx}{dt} + (
  • #1
mr_coffee
1,629
1
Hello everyone...
I'm very confused...
i'm suppose to find
dz/dt and dw/dt
but for some of the questions there is no w variable! so what do u put for dw/dt?! Also i have the following:
w = xy + yz^2; x = e^t; y = e^t*sint; z = e^t*cost;
so I'm trying to find dz/dt and dw/dt;
dz/dt = dz/dx * dx/dt + dz/dy * dy/dt + dz/dw * dw/dt;
but when i try dz/dx * dx/dt i need to first take the partial derivative of z with respect to x, but as you can see, z has no x variable! so what do i do about that? THanks!
 
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  • #2
how do you get dz/dt = dz/dx * dx/dt +dz/dy * dy/dt + dz/dw * dw/dt

dz/dt[e^t * cos t] = cos t * e^t - sint * e^t

there is no dx

as far as dw/dt

it would be

dw/dx * dx/dt + dw/dy * dy/dt + dw/dz + dz/dt
 
  • #3
mr_coffee:

You need to keep track of what is a function of what.

You have:

[tex]w = xy + yz^2; x = e^t; y = e^t \sin t; z = e^t \cos t[/tex]

So, you have:

[tex]w = w(x,y,z); x = x(t); y = y(t); z=z(t)[/tex]

If you want to find dz/dt, it's just a simple derivative, since z is only a function of t, and not of x or y or w.

To find dw/dt you need the chain rule:

[tex]\frac{dw}{dt} = \frac{\partial w}{\partial x} \frac{dx}{dt} + \frac{\partial w}{\partial y} \frac{dy}{dt} + \frac{\partial w}{\partial z} \frac{dz}{dt}[/tex]
 

FAQ: Chain rule confusion partial derivatives

1. What is the chain rule?

The chain rule is a mathematical rule used to find the derivative of a composition of functions. It states that the derivative of the outer function multiplied by the derivative of the inner function gives the derivative of the composite function.

2. How is the chain rule applied in partial derivatives?

In partial derivatives, the chain rule is used to find the derivative of a function with respect to one of its variables while holding the other variables constant. This is done by treating the other variables as constants and applying the chain rule as usual.

3. Why is the chain rule sometimes confusing?

The chain rule can be confusing because it requires a good understanding of basic calculus concepts and the ability to recognize and apply it correctly in different situations. It also involves multiple steps and can become more complex when dealing with higher order derivatives.

4. What are some common mistakes when using the chain rule in partial derivatives?

Some common mistakes when using the chain rule in partial derivatives include forgetting to treat the other variables as constants, incorrectly applying the chain rule formula, and not simplifying the final expression properly.

5. How can I improve my understanding and application of the chain rule in partial derivatives?

To improve your understanding of the chain rule in partial derivatives, it is important to review and practice basic calculus concepts, familiarize yourself with different examples and applications of the rule, and seek help from a tutor or teacher if necessary. Additionally, breaking down the steps and simplifying the expressions can help in avoiding mistakes.

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