Page:Calculus Made Easy.pdf/194

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

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

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

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

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

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

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

(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 ${\displaystyle x}$, or for very large values of ${\displaystyle x}$, or for values of ${\displaystyle x}$ in the neighbourhood of ${\displaystyle x=30}$.

(12) Differentiate the following:

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

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

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