 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2016-resit/5/solution

x \mapsto (x, y (x))

, so the chain rule gives

\frac{d u}{d x} = (\frac{\partial u}{\partial x})_{y} + (\frac{\partial u}{\partial y})_{x} \frac{d y}{d x} .

For the stated functions,

u_{x} = 6 x

u_{y} = 2 y

and

y^{'} = cos x

, hence

\frac{d u}{d x} = 6 x + 2 sin x cos x = 6 x + sin 2 x .

u (x, sin x) = 3 x^{2} + sin^{2} x

gives the same result.

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