Techniques of Differentiation - Second and Higher Derivatives
Techniques of Differentiation - Second and Higher Derivatives
TGM Original Questions
A curve has the equation \(y = \frac{3x - 2}{x^2 + 4}\).
- Obtain expressions for \(\frac{\mathrm{d}y}{\mathrm{d}x}\) and \(\frac{\mathrm{d}^2y}{\mathrm{d}x^2}\) in the simplest form.
- Hence find, respectively, the range of values of \(x\) for which \(y\) is increasing and decreasing with respect to \(x\).
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Answer:(i) \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{-3x^2+4x+12}{(x^2+4)^2},\ \dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}=\dfrac{2(3x^3-6x^2-36x+8)}{(x^2+4)^3}\) (ii) Increasing for \(\dfrac{2-2\sqrt{10}}{3}<x<\dfrac{2+2\sqrt{10}}{3}\); decreasing for \(x < \dfrac{2-2\sqrt{10}}{3}, x > \dfrac{2+2\sqrt{10}}{3}\)
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