Page:Calculus Made Easy.pdf/174

Differentiate

(6) $y=\log _{\epsilon }x^{n}$.

(7) $y=3\epsilon ^{-{\frac {x}{x-1}}}$.

(8) $y=(3x^{2}+1)\epsilon ^{-5x}$.

(9) $y=\log _{\epsilon }(x^{a}+a)$.

(10) $y=(3x^{2}-1)({\sqrt {x}}+1)$.

(11) $y={\dfrac {\log _{\epsilon }(x+3)}{x+3}}$.

(12) $y=a^{x}\times x^{a}$.

(13) It was shown by Lord Kelvin that the speed of signalling through a submarine cable depends on the value of the ratio of the external diameter of the core to the diameter of the enclosed copper wire. If this ratio is called $y$, then the number of signals $s$ that can be sent per minute can be expressed by the formula

$s=ay^{2}\log _{\epsilon }{\frac {1}{y}}$;

where $a$ is a constant depending on the length and the quality of the materials. Show that if these are given, $s$ will be a maximum if $y=1\div {\sqrt {\epsilon }}$.

(15) Differentiate $y=\log _{\epsilon }(ax\epsilon ^{x})$.

(16) Differentiate $y=(\log _{\epsilon }ax)^{3}$.

 

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The Logarithmic Curve.

Let us return to the curve which has its successive ordinates in geometrical progression, such as that represented by the equation $y=bp^{x}$.

We can see, by putting $x=0$, that $b$ is the initial height of $y$.

Then when

$x=1,\;y=bp;\;x=2,y=bp^{2};\;x=3,y=bp^{3},\;$ etc.