A student’s partial solution to a higher-order derivative problem is reproduced below

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

A student’s partial solution to a higher-order derivative problem is reproduced below. "Given h(t)=ln⁡(t2+1)h(t)=\ln(t^2+1), we have h′(t)=2tt2+1,h′′(t)=2t2+1−4t2(t2+1)2,h'(t)=\frac{2t}{t^2+1}, \quad h''(t)=\frac{2}{t^2+1}-\frac{4t^2}{(t^2+1)^2}, so d2hdt2(t)=h′′(t),    and    h(3)(t)=d3hdt2.\frac{d^2h}{dt^2}(t)=h''(t), \;\; \text{and} \;\; h^{(3)}(t)=\frac{d^3h}{dt^2}." (a) Identify every instance of incorrect or ambiguous derivative notation in the student’s work. List each notation error separately. (b) Rewrite the student’s derivative statements from the quoted passage using correct and unambiguous notation. Ensure that each derivative is expressed consistently at the same level of generality. (c) Justify why the corrected notations in part (b) accurately convey the order of each derivative. Your justification should reference the defining features of the notations you use.

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