Calculating Higher-Order Derivatives AB & BC

A higher-order derivative is produced by differentiating again, and the object being differentiated on the second pass is $f'$, a function in its own right. Its structure is often more complicated than the original's, so the second pass frequently requires more rules than the first: a power rule can produce a product, and a composition contributes a fresh inner factor every time it is differentiated. The second derivative of $\sin(kx)$ carries $k^2$ for exactly that reason.

The characteristic error is treating the second pass as mechanical, differentiating $f'$ as if it were a simple power and dropping the chain, product, or implicit rules that its structure still demands. Implicit second derivatives add a second step that also gets skipped: after differentiating $\frac{dy}{dx}$ with the quotient rule, the expression still contains $\frac{dy}{dx}$, and the first derivative has to be substituted back in before the answer is complete.