 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-1/18/a/ii/solution

At fixed

t

, the argument is linear in

x

, with

\partial s / \partial x = b^{a} t^{a}

. Each differentiation therefore contributes this same constant factor, while the outside factor

t^{a}

remains unchanged. Thus

\frac{\partial ^{3} u}{\partial x ^{3}} = t^{a} (b^{a} t^{a})^{3} y^{(3)} (s) = b^{3 a} t^{4 a} y^{(3)} (s) .

Here

y^{(3)}

denotes the third derivative of the single-variable profile, not a partial derivative of

u

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