Past exam of the mathematical methods II course of the University of Oxford / 2016 resit / 5

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- Solution 5

Solution

The total derivative follows the curve

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 .

Direct differentiation of

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

gives the same result.

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