What sort of applications are there for $(dy/dx) (dx/dy) = 1$?

Question

A calculus exercise that I just did concludes that $$\frac{dy}{dx} \frac{dx}{dy} = 1.$$ The exercise remarks that this is true in general. In what sort of situation can I use this? Can you show an exercise where I can take advantage of a result like that? (A typical exercise or practical application would be great.)

Answer 1

It helps find the derivatives of inverse functions.

Example: Find the derivative of $y = \arcsin(x)$ ($x \in [-1,1]$, $y \in [-\frac{\pi}{2}, \frac{\pi}{2}]$), knowing that $\frac{d}{dx} \sin(x) = \cos(x)$.

By the definition of an inverse function:

$$\sin(y) = x$$

Differentiating wrt $y$ gives:

$$\cos(y) = \frac{dx}{dy}$$

Taking the reciprocal (which your question acknowledges that you can do) gives:

$$\frac{dy}{dx} = \frac{1}{\cos y}$$ $$\frac{dy}{dx} = \frac{1}{\cos(\arcsin(x))}$$

From the Pythagorean identity $\cos^2 \theta + \sin^2 \theta = 1$, with $x = \sin \theta$, this becomes:

$$\frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}}$$

Answer 2

As an example of where it can be used, suppose $y={\rm e}^{\theta}$ where $x=f(\theta)$ and we wish to obtain the derivative of $y\,$ with respect to $x.$ We get $$\dfrac{{\rm d}y}{{\rm d}x}={\rm e}^{\theta}\,\dfrac{{\rm d}\theta}{{\rm d}x}.$$ The derivative of $\theta=f^{-1}(x)$ with respect to $x$ may be difficult to obtain, but this can easily be avoided by noting that it can be evaluated simply as $$ \dfrac{{\rm d}\theta}{{\rm d}x}=\left(\!\dfrac{{\rm d}x}{{\rm d}\theta}\! \right)^{\!\!-1}, $$so that the derivative ${\rm d}y/{\rm d}x$ is easily obtained as $$\dfrac{{\rm d}y}{{\rm d}x}= {\rm e}^{\theta} \left(\!\dfrac{{\rm d}f(\theta)}{{\rm d}\theta}\!\right)^{\!\!-1}, $$without having to obtain the inverse $\theta=f^{-1}(x).$