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If $$\frac{dy}{dx} = \left(\frac{dy}{du}\right)\left(\frac{du}{dx}\right),$$ will the inverse of $dy/dx$ flip the other derivatives? For instance, will $$\frac{dx}{dy} = \left(\frac{du}{dy}\right)\left(\frac{dx}{du}\right)?$$ Why/why not?

soupless
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seb aye
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  • Why would you need to flip them, are you carrying out an implicit derivative of a function of a function? –  Jun 15 '21 at 08:01

1 Answers1

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I assume that you are using the word inverse as the reciprocal of a term. Then, yes, inverting $\frac{dy}{dx}$ also inverts the other terms.

As you stated, $$\frac{dy}{dx} = \left(\frac{dy}{du}\right)\left(\frac{du}{dx}\right).$$

Then, \begin{align*} \frac{dx}{dy} &= \left(\frac{du}{dy}\right)\left(\frac{dx}{du}\right) \\ \frac{dx}{dy} &= \left(\frac{dx}{du}\right)\left(\frac{du}{dy}\right) \end{align*}

where the last equation is when we apply the chain rule for $\frac{dx}{dy}$. Hence, we can somehow, say that inverting $\frac{dy}{dx}$ also inverts the terms when the chain rule applied.

But remember that this is just an abuse of notation. About this, see this answer.

soupless
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