Techniques of Differentiation - Derivatives of Natural Logarithmic Functions
Techniques of Differentiation - Derivatives of Natural Logarithmic Functions
TGM Original Questions
IB AA SL: core and optional content
Core: differentiate ln(x²+3). Optional: partial-fraction decomposition and the dependent integral.
- Express \(\frac{27x^2-6x+9}{(4x-1)(x^2+3)}\) in partial fractions.
- Differentiate \(\ln(x^2+3)\) with respect of \(x\).
- Using the results of (a) and (b), find \(\displaystyle\displaystyle \displaystyle\int \frac{27x^2-6x+9}{(4x-1)(x^2+3)} \, \mathrm{d}x\).
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Answer:(a) \(\frac3{4x-1}+\frac{6x}{x^2+3}\). (b) \(\frac{2x}{x^2+3}\). (c) \(\frac34\ln|4x-1|+3\ln(x^2+3)+C\).
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