Q.If $y = x^{4}$, find the second-order derivative $\dfrac{d^{2}y}{dx^{2}}$.

Differentiate once, then differentiate the result again. …

First derivative: $\dfrac{dy}{dx}=\dfrac{d}{dx}(x^4)=4x^3$.

Second derivative — differentiate $4x^3$ again:

$\frac{d^2y}{dx^2}=\frac{d}{dx}(4x^3)=4\cdot3x^2=12x^2.$ …

$\frac{d^2y}{dx^2}$ means differentiate twice, not square the first derivative. Compute $4x^3$ …