Further Applications of Differentiation (HSC SSCE Mathematics Advanced): Flashcards

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Further Applications of Differentiation
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ddx(log⁡ex)\frac{d}{dx}(\log_e x)

1x\frac{1}{x}

Domain of log⁡ex\log_e x

x>0x > 0

Normal gradient when tangent gradient is mm

−1m-\frac{1}{m} (negative reciprocal)

ddx(xlog⁡ex)\frac{d}{dx}(x \log_e x)

log⁡ex+1\log_e x + 1

d2dx2(xlog⁡ex)\frac{d^2}{dx^2}(x \log_e x)

1x\frac{1}{x}

Zero of xlog⁡exx \log_e x

x=1x = 1

lim⁡x→0+xlog⁡ex\lim_{x \to 0^+} x \log_e x

00

lim⁡x→∞log⁡exx\lim_{x \to \infty} \frac{\log_e x}{x}

00

Meaning of 'xx dominates log⁡ex\log_e x'

xx grows/shrinks faster than log⁡ex\log_e x

If f′′(x)>0f''(x) > 0 at stationary point, nature is

Minimum

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