Chain rule for higher-order derivatives

Question

While Studying Chain rule In my Calculus Book It was written as :-

$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$

But, In the note it was mentioned that :-

$\frac{d^2y}{dx^2} \ne \frac{d^2y}{du^2}\frac{d^2u}{dx^2}$

If the Chain rule is not valid for higher-order derivatives in this form Then what is the correct rule?

Answer 1

The chain rule is basically $$ (f(g(x)))'=f'(g(x))g'(x)$$

Thus if we take the second derivative we have to use product rule to get

$$ (f(g(x)))''=f''(g(x))g'^2(x)+ f'(g(x))g''(x)$$

$$\frac{d^2y}{dx^2} = \frac{d^2y}{du^2} (\frac{du}{dx})^2 +\frac{dy}{du} \frac{d^2u}{dx^2} $$