If x = 3z + 1 and y = f(x), then prove that 9(d²y)/(dx²) = (d²y)/(dz²).

Since $x = 3z + 1$, $\frac{dx}{dz} = 3$. By the chain rule, $\frac{dy}{dz} = \frac{dy}{dx}\cdot\frac{dx}{dz} = 3\frac{dy}{dx}$. Differentiating again with respect to $z$: $\frac{d^2y}{dz^2} = 3\frac{d}{dz}\left(\frac{dy}{dx}\right) = 3\frac{d}{dx}\left(\frac{dy}{dx}\right)\cdot\frac{dx}{dz} = 3\cdot 3\frac{d^2y}{dx^2}$. So $\frac{d^2y}{dz^2} = 9\frac{d^2y}{dx^2}$. Hence proved.