# What to use when reverse chain rule doesnt work?

• highc1157
In summary, The conversation discusses using the reverse chain rule to solve an equation that cannot be solved using exact solutions. It is mentioned that any equation can be made exact by finding the correct coordinates. The equation is then rewritten in terms of these coordinates as (-y-sqrt(-2(-x2/2)))d(-x2/2)-dy.
highc1157
Hi there,

My equation to solve is (xy+(x^2))dx + (-1)dy=0

For method of exact solutions, the partials are not equal to each other so I cannot use
exact solutions (reverse chain rule)

I don't know how to solve this

What is reverse chain rule?
Any equation is exact in some coordinates so find them.
you stated (correctly)
(xy+x2)y - (-1)x
is not zero but we can use the fact that
(((xy+x2)y -(-1)x)/(-1))y=0
to conclude that the equation is exact in
∫((xy+x2)y - (-1)x)/(-1)∂x
and y since (xy+(x2))dx + (-1)dy can be written
(-y-sqrt(-2(-x2/2)))d(-x2/2)-dy

## What is the reverse chain rule?

The reverse chain rule, also known as the chain rule for inverse functions, is a mathematical rule used to find the derivative of a function that is composed of two or more functions. It is the opposite of the regular chain rule and is used when the dependent and independent variables are switched.

## When does the reverse chain rule not work?

The reverse chain rule does not work when the function is not a composite function, meaning it cannot be broken down into two or more functions. It also does not work when the dependent and independent variables are not switched, or when the function is not differentiable at a certain point.

## What are some alternative methods when the reverse chain rule doesn't work?

One alternative method is to use the quotient rule, which is used when finding the derivative of a function that is in the form of a fraction. Another method is to use the product rule, which is used when taking the derivative of a product of two functions. In some cases, it may be necessary to use numerical or graphical methods to approximate the derivative.

## How can I determine when to use the reverse chain rule?

The reverse chain rule should be used when the dependent and independent variables are switched, and the function can be broken down into two or more functions. It should also be used when the original function is not differentiable at a certain point. It is important to carefully analyze the given function and determine if the reverse chain rule is applicable.

## Are there any common mistakes when using the reverse chain rule?

Yes, some common mistakes when using the reverse chain rule include forgetting to switch the dependent and independent variables, not breaking down the function into two or more functions, and not applying the rule correctly. It is important to double check the steps and calculations to avoid any errors.

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