Techniques of Differentiation - Derivatives of Natural Logarithmic Functions
Techniques of Differentiation - Derivatives of Natural Logarithmic Functions
IB AA SL: core and optional content
Core: differentiate the logarithm on its real domain. Optional: partial-fraction decomposition and the dependent integral.
When differentiating the stated real logarithm, use only inputs for which its argument is positive. The corresponding antiderivative may use an absolute value to cover other connected intervals where the rational integrand is defined.
- Express \(\frac{-x^2 + 2x - 2}{(1-x)(x^2 - 3x + 1)}\) in partial fractions.
- Differentiate \(\ln(x^2 - 3x + 1)\) with respect to \(x\).
- Using your results in part (a) and (b), find \(\displaystyle\displaystyle \displaystyle\int \frac{\frac{1}{2}x^2 - x + 1}{(1-x)(x^2 - 3x + 1)} \, \mathrm{d}x\).
Solution:
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Answer:(a) \(\frac1{1-x}+\frac{2x-3}{x^2-3x+1}\). (b) \(\frac{2x-3}{x^2-3x+1}\), where \(x^2-3x+1>0\). (c) \(\frac12\ln|1-x|-\frac12\ln|x^2-3x+1|+C\), away from all denominator zeros.
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