How to apply chain rule to:$\frac{d}{dx} \Big( \frac{dy}{dx} \Big)$?

Question

How do I apply chain rule to the following: $$\frac{d}{dx} \Big( \frac{dy}{dx} \Big)$$

Where $$\Big( \frac{dy}{dx} \Big) = \Bigg(\frac{\frac{dy}{dt}}{\frac{dx}{dt}} \Bigg)$$

I don't see the relationship between the bottom $dx$ and the $dy$ in the numerator. My textbook states that:

$$\frac{d}{dx} \Big( \frac{dy}{dx} \Big) = \frac{\frac{d}{dt} \times \big( \frac{dy}{dx} \big)}{\frac{dx}{dt}}$$

without explanation.

Any hints or help on the details of this proccess will be greatly appreciated. Cheers!

Answer 1

The chain rule says

$$\frac{d\phi}{dx}\frac{dx}{dt}=\frac{d\phi}{dt}.$$

Divide both sides by $\frac{dx}{dt}$ and set $\phi=\frac{dy}{dx}$ to get the desired formula.

Better, just use the original formula

$$\frac{d}{dx}=\left(\frac{dx}{dt}\right)^{-1}\frac{d}{dt}$$

(as an equality of operators) on $\frac{dy}{dx}$ instead of $y$.

Answer 2

Let $y^{\prime}=\frac{dy}{dx}$. By the Chain Rule,

$\displaystyle\frac{dy^{\prime}}{dx}=\frac{\frac{dy^{\prime}}{dt}}{\frac{dx}{dt}}=\frac{\frac{d}{dt}(\frac{dy/dt}{dx/dt})}{\frac{dx}{dt}}=\frac{\frac{dx}{dt}\cdot\frac{d^2y}{dt^2}-\frac{dy}{dt}\cdot\frac{d^2x}{dt^2}}{(\frac{dx}{dt})^3}$ (using the Quotient Rule to simplify the top).