Chain rule partial derivative

In summary, the chain rule for partial derivatives is a mathematical rule used to find the partial derivative of a composite function. It is used when taking the partial derivative of a function that is composed of two or more functions, and is different from the chain rule for ordinary derivatives. It is always applicable as long as the functions involved are differentiable, and can be used in various real-world applications such as in physics, engineering, economics, and finance.
  • #1
wololo
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Homework Statement


Capture.PNG


Homework Equations


Chain rule, partial derivation

The Attempt at a Solution


dv/dt=dv/dx*dx/dt+dv/dy*dy/dt
dx/dt=-4t -> evaluate at (1,1) =-4
dv/dt=-4dv/dx+4(-2)
dv/dt=-4dv/dx-8

How can I find the missing dv/dx in order to get a value for dv/dt? Thanks!
 
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  • #2
Did you expand the other derivatives to see if you get additional constraints from them?
 

1. What is the chain rule for partial derivatives?

The chain rule for partial derivatives is a mathematical rule used to find the partial derivative of a composite function. It states that the partial derivative of a composite function is equal to the partial derivative of the outer function multiplied by the partial derivative of the inner function.

2. When is the chain rule for partial derivatives used?

The chain rule for partial derivatives is used when taking the partial derivative of a function that is composed of two or more functions. This can commonly occur in multivariable calculus or in the field of physics when dealing with complex systems.

3. How is the chain rule for partial derivatives different from the chain rule for ordinary derivatives?

The chain rule for partial derivatives is different from the chain rule for ordinary derivatives because it deals with functions of multiple variables. In the chain rule for ordinary derivatives, the derivative of a composite function is found by multiplying the derivatives of each individual function. However, in the chain rule for partial derivatives, the partial derivatives of each individual function are multiplied together and then multiplied by the partial derivative of the outer function.

4. Is the chain rule for partial derivatives always applicable?

The chain rule for partial derivatives is always applicable as long as the functions involved are differentiable. This means that the functions must have a defined partial derivative at the given point.

5. How do I use the chain rule for partial derivatives in a real-world application?

The chain rule for partial derivatives can be used in various real-world applications, such as in physics and engineering. For example, it can be used to calculate the rate of change of a quantity with respect to multiple variables, or to find the gradient of a scalar field. It is also used in economics and finance to analyze the relationship between multiple variables in a system.

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