How Can I Prove the Reciprocal Derivative Identity?

In summary, the conversation discusses how to prove the identity \frac{{dy}}{{dx}}\frac{{dx}}{{dy}} = 1 and the use of the chain rule in solving it. The individual asking the question has been able to show that the identity holds true for some examples but is seeking a solid proof. The expert then provides a solution using the chain rule.
  • #1
danago
Gold Member
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4
Hi. I was just wondering, how can i prove the following identity:

[tex]
\frac{{dy}}{{dx}}\frac{{dx}}{{dy}} = 1
[/tex]

Its nothing that I am required to know, but i was just curious, so for all i know, it may be way out of anything that i can mathematically comprehend.

The best I've been able to do is show that it holds true for some examples that I've tried, but no solid proof.

Thanks in adavnce,
Dan.
 
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  • #2
It should be easy using the chain rule. If y= f(x) and f is invertible, then
x= f-1(y), so that f-1(f(x))= x. Differentiating both sides of that with respect to x,
[tex]\frac{df^{-1}(y)}{dy}\frac{dy}{dx}= 1[/itex]
Where I have 'let' y= f(x). Since f-1(y)= x, that is
[tex]\frac{dx}{dy}\frac{dy}{dx}= 1[/itex]
 
  • #3
Ah, easy. Thanks very much for that :smile:
 

FAQ: How Can I Prove the Reciprocal Derivative Identity?

What is the Reciprocal Derivative Identity?

The Reciprocal Derivative Identity is a mathematical rule that states the derivative of the reciprocal of a function is equal to the negative value of the reciprocal of the derivative of the original function.

Why is the Reciprocal Derivative Identity important?

The Reciprocal Derivative Identity is important because it allows us to find the derivative of a reciprocal function without having to use the quotient rule. It also helps us to simplify complex derivatives and solve problems more efficiently.

How is the Reciprocal Derivative Identity used in real life?

The Reciprocal Derivative Identity is used in many fields of science and engineering, such as physics, chemistry, and economics. It is particularly useful in calculating rates of change, optimization problems, and analyzing motion and growth.

What is the difference between the Reciprocal Derivative Identity and the Quotient Rule?

The Reciprocal Derivative Identity and the Quotient Rule are two different methods for finding the derivative of a reciprocal function. The Reciprocal Derivative Identity is a shortcut that only applies to reciprocal functions, while the Quotient Rule can be used for any type of quotient function.

How can the Reciprocal Derivative Identity be proven?

The Reciprocal Derivative Identity can be proven using the Chain Rule and the Power Rule for derivatives. By applying these rules to the reciprocal function, we can show that the derivative of the reciprocal equals the negative reciprocal of the derivative of the original function.

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