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Techniques of Differentiation - Second and Higher Derivatives

Techniques of Differentiation - Second and Higher Derivatives

A curve has the equation \(\mathrm{f}(x)=\frac{2x^2}{(x+2)}\),

  1. Obtain expressions for \(\mathrm{f}'(x)\) and \(\mathrm{f}''(x)\). Simplify your answers.
  2. Hence find the range of values of \(x\) for which \(\mathrm{f}(x)\) is
    1. an increasing function and
    2. a decreasing function.

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Answer:(a) \(\mathrm f'(x)=\frac{2x(x+4)}{(x+2)^2}\), \(\mathrm f''(x)=\frac{16}{(x+2)^3}\) (b)(i) \(x<-4\) or \(x>0\) (ii) \(-4<x<-2\) or \(-2<x<0\)

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