Page:Calculus Made Easy.pdf/194

(4) If $y=\sin a^{x}$, find ${\dfrac {dy}{dx}}$.

(5) Differentiate $y=\log _{\epsilon }\cos x$.

(6) Differentiate $y=18.2\sin(x+26^{\circ })$.

(7) Plot the curve $y=100\sin(\theta -15^{\circ })$; and show that the slope of the curve at $\theta =75^{\circ }$ is half the maximum slope.

(8) If $y=\sin \theta \cdot \sin 2\theta$, find ${\dfrac {dy}{d\theta }}$.

(9) If $y=a\cdot \tan ^{m}(\theta ^{n})$, find the differential coefficient of $y$ with respect to $\theta$.

(10) Differentiate $y=\epsilon ^{x}\sin ^{2}x$.

(11) Differentiate the three equations of Exercises XIII. (p. 163), No. 4, and compare their differential coefficients, as to whether they are equal, or nearly equal, for very small values of $x$, or for very large values of $x$, or for values of $x$ in the neighbourhood of $x=30$.

(12) Differentiate the following:

${\begin{aligned}{\text{(i)}}\quad y&=\sec x.\\{\text{(ii)}}\quad y&=\arccos x.\\{\text{(iii)}}\quad y&=\arctan x.\\{\text{(iv)}}\quad y&={\text{arcsec}}x.\\{\text{(v)}}\quad y&=\tan x\times {\sqrt {3\sec x}}.&&\end{aligned}}$

(13) Differentiate $y=\sin(2\theta +3)^{2.3}$.

(14) Differentiate $y=\theta ^{3}+3\sin(\theta +3)-3^{\sin \theta }-3^{\theta }$.

(15) Find the maximum or minimum of $y=\theta \cos \theta$