Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 1

Table of contents
- a 1
  - Solution a
- b 1
  - Solution b

a

Solution

Write the rational expression as

(x^{2} + 4)^{- 1}

and apply the chain rule:

\frac{d}{d x} \frac{1}{x ^{2} + 4} = - \frac{2 x}{( x ^{2} + 4 ) ^{2}} .

The denominator is positive for every real

x

, so the derivative is defined throughout the real line.

b

Solution

The outer exponential function differentiates to itself, and the inner sine has derivative cosine. The chain rule gives

\frac{d}{d x} e^{s i n x} = e^{s i n x} cos x .

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