Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 1 / 8

Table of contents
- a 8
  - Solution a
- b 8
  - Solution b

a

Solution

For

a > 0

, write

a^{x} = e^{x l n a}

and apply the chain rule:

\frac{d}{d x} a^{x} = (ln a) a^{x} .

This also covers

a = 1

, for which both sides are zero.

b

Solution

Differentiate the outer sine, then the inner cosine. The chain rule gives

\frac{d}{d x} sin (cos x) = - sin x cos (cos x) .

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