(a) Find, from first principles, the derivative of (f(x) = (2x + 3)^{2}).
(b) Evaluate : (int_{1} ^{2} frac{(x + 1)(x^{2} – 2x + 2)}{x^{2}} mathrm {d} x)
Explanation
(a) The formula for the first principles method of calculating derivatives is
(frac{mathrm d y}{mathrm d x} = lim limits_{Delta x to 0} frac{f(x + Delta x) – f(x)}{Delta x})
(f(x) = (2x + 3)^{2} = 4x^{2} + 12x + 9)
(f(x + Delta x) = 4(x + Delta x)^{2} + 12(x + Delta x) + 9)
= (4x^{2} + 8x Delta x + 4 (Delta x)^{2} + 12x + 12Delta x + 9)
(lim limits_{Delta x to 0} frac{(4x^{2} + 8x Delta x + 4(Delta x)^{2} + 12x + 12Delta x + 9) – (4x^{2} + 12x + 9)}{Delta x})
= (lim limits_{Delta x to 0} frac{4(Delta x)^{2} + (8x + 12)Delta x}{Delta x})
= (lim limits_{Delta x to 0} 4 Delta x + (8x + 12))
(frac{mathrm d y}{mathrm d x} = 8x + 12)
(b) (int_{1} ^{2} frac{(x + 1)(x^{2} – 2x + 2)}{x^{2}} mathrm {d} x)
= (int_{1} ^{2} frac{x^{3} – 2x^{2} +2x + x^{2} – 2x + 2}{x^{2}} mathrm {d} x)
= (int_{1} ^{2} frac{x^{3} – x^{2} + 2}{x^{2}} mathrm {d} x)
= (int_{1} ^{2} (x – 1 + 2x^{2}) mathrm {d} x)
= ([frac{x^{2}}{2} – x – frac{2}{x}]_{1} ^{2} )
= ((frac{2^{2}}{2} – 2 – frac{2}{2}) – (frac{1^{2}}{2} – 1 – frac{2}{1}))
= ((2 – 2 – 1) – (frac{1}{2} – 1 – frac{2}{1}))
= (-1 – (-frac{5}{2}) = frac{3}{2})