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

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 $$y=(\log_\epsilon ax)^3$$.

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