Partial Derivatives (Chain Rule)

In summary, the given equations for x, y, and z can be used to find the partial derivatives of w with respect to r and theta, which are zcos(theta) + 2ysin(theta) and -rzsin(theta) + 2rycos(theta) respectively. It is recommended to replace x and y in the right-hand side of each equality with their new definitions in order to accurately reflect the change of variables in the differentiations. Additionally, when working with cylindrical coordinates, it is easier to convert all variables to cylindrical coordinates before finding the partial derivatives.
  • #1
DeadxBunny
30
0
Original question:

Let w = y^2 + xz. If x = rcos(theta), y = rsin(theta), and z = z, find (partial w)/(partial r) and (partial w)/(partial theta).

Could someone please check my answers?

(partial w)/(partial r) = zcos(theta) + 2ysin(theta)

(partial w)/(partial theta) = -rzsin(theta) + 2rycos(theta)

Thank you!
 
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  • #2
The results are outstanding.They would be perfect if u replaced in the RHS of each equality "x" and "y" through their new definitions.Only that way it could be said u made a change of variable and in the differentiations u took it into consideration.
 
  • #3
Looks correct. Usually when dealing with cylindrical coors, one converts all variables over. Meaning, it is a bit easier to convert all x's and y's to cylindrical coors then do the partial derivatives.
 

1. What is the chain rule in partial derivatives?

The chain rule in partial derivatives is a method used to find the derivative of a function with multiple variables. It states that the derivative of a function of multiple variables can be found by multiplying the partial derivatives of each variable with respect to an intermediate variable.

2. How do you apply the chain rule in partial derivatives?

To apply the chain rule in partial derivatives, you first need to identify the variables in the function and the intermediate variable. Then, you take the partial derivative of the function with respect to the intermediate variable. Finally, you multiply this derivative with the partial derivative of the intermediate variable with respect to the original variable.

3. Can the chain rule be applied to any type of function?

Yes, the chain rule can be applied to any type of function, as long as it involves multiple variables. It is commonly used in functions involving trigonometric, logarithmic, and exponential functions.

4. What is the difference between the chain rule in partial derivatives and the chain rule in single variable calculus?

The chain rule in single variable calculus is used to find the derivative of a function with one independent variable. It involves taking the derivative of the outer function and multiplying it by the derivative of the inner function. On the other hand, the chain rule in partial derivatives takes into account multiple variables and involves multiplying partial derivatives instead of single variable derivatives.

5. Why is the chain rule important in multivariable calculus?

The chain rule is important in multivariable calculus because it allows us to find the derivative of a function with multiple variables, which is necessary for understanding the behavior of complex systems. It also helps in optimizing functions with multiple variables, which is crucial in many real-world applications such as economics, engineering, and physics.

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