 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2013/discrete-mathematics-probability-and-continuous-mathematics/4/b/solution

The multivariable chain rule follows the change of both coordinates along the parametric curve:

\frac{df}{d s} = f_{x} \frac{d x}{d s} + f_{y} \frac{d y}{d s}, \frac{d x}{d s} = 2 s, \frac{d y}{d s} = e^{s} .

Substituting the partial derivatives from part (a), with

x = 1 + s^{2}

and

y = e^{s}

, yields

\frac{df}{d s} = 4 s (1 + s^{2}) (e^{4 s} + e^{2 s} + 1) + 4 (1 + s^{2})^{2} e^{4 s} + 3 e^{3 s} + 2 (1 + s^{2})^{2} e^{2 s} - 3 e^{s} .

The first term comes from changing

x

, and the remaining terms from changing

y

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