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Techniques of Differentiation - Second and Higher Derivatives

Techniques of Differentiation - Second and Higher Derivatives

A curve has the equation \(\mathrm{f}(x)=\frac{3x^2}{2x-1}\),

  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) \(\mathrm f'(x)=\frac{6x(x-1)}{(2x-1)^2}\), \(\mathrm f''(x)=\frac6{(2x-1)^3}\) (b)(i) \(x<0\) or \(x>1\) (ii) \(0<x<\frac12\) or \(\frac12<x<1\)

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