Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2024 / Paper 1 / 19 / a

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- Solution a

Solution

Apply the Leibniz rule for the

n

th derivative of a product. Only the first three terms survive, because all derivatives of

x^{2}

of order greater than two vanish and every derivative of

e^{x}

is

e^{x}

:

\frac{d ^{n}}{d x ^{n}} (x^{2} e^{x}) = e^{x} [(0 n) x^{2} + (1 n) (2 x) + (2 n) (2)] = e^{x} [x^{2} + 2 n x + n (n - 1)] .

At

n = 100

this gives

\frac{d ^{100}}{d x ^{100}} (x^{2} e^{x}) = e^{x} (x^{2} + 200 x + 9900) .

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