PART 15：例題-對數微分法

假設 \(y = {(2x + 1)^5}(3x - 4)({x^2} + x + 1)\) ，求 \(\left. {\frac{{dy}}{{dx}}} \right|_{x = 0}^{}\)

SOL:

(1)兩邊取對數：

\(\ln y = 5\ln (2x + 1) + \ln (3x - 4) + \ln ({x^2} + x + 1)\)

(2) 兩邊微分：

\(\frac{1}{y}y' = \frac{{10}}{{2x + 1}} + \frac{3}{{3x - 4}} + \frac{{2x + 1}}{{{x^2} + x + 1}}\)

(3)解出 \(y'\)

\(y' = {(2x + 1)^5}(3x - 4)({x^2} + x + 1)\left[ {\frac{{10}}{{2x + 1}} + \frac{3}{{3x - 4}} + \frac{{2x + 1}}{{{x^2} + x + 1}}} \right]\)

　 \( = \left( { - 4} \right)\left[ {10 - \frac{3}{4} + 1} \right] =  - 41\)