Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/126

Use Leibnitz's Formula in the next four examples:
24. $y=x^{2}a^{x}$.	${\frac {d^{n}y}{dx^{n}}}=a^{x}(\log a)^{n-2}[(x\log a+n)^{2}-n]$
25. $y=e^{x}x$.	${\frac {d^{n}y}{dx^{n}}}=e^{x}(x+n)$.
26. $f(x)=e^{x}\sin x$.	$f^{(n)}(x)=({\sqrt {2}})^{n}e^{x}sin\left(x+{\frac {n\pi }{4}}\right)$.
27. $f(\theta )=\cos a\theta \cos b\theta$.	$f^{n}(\theta )={\frac {(a+b)^{n}}{2}}\cos \left[(a+b)\theta +{\frac {n\pi }{2}}\right]$ $+{\frac {(a-b)^{n}}{2}}\cos \left[(a-b)\theta +{\frac {n\pi }{2}}\right]$.
28. Show that the formulas for acceleration, (14), (15), §71, may be written
$\alpha ={\frac {d^{2}s}{dt^{2}}},\alpha _{x}={\frac {d^{2}x}{dt^{2}}},\alpha _{y}={\frac {d^{2}y}{dt^{2}}}$.
29. $y^{2}=4ax$.	${\frac {d^{2}y}{dx^{2}}}=-{\frac {4a^{2}}{y^{3}}}$.
30. $b^{2}x^{2}+a^{2}Y^{2}=a^{2}b^{2}$.	${\frac {d^{2}y}{dx^{2}}}=-{\frac {b^{4}}{a^{2}y^{3}}};{\frac {d^{3}y}{dx^{2}}}=-{\frac {3b^{6}x}{a^{4}y^{5}}}$.
31. $x^{2}+y^{2}=r^{2}$.	${\frac {d^{2}y}{dx^{2}}}=-{\frac {r^{2}}{y^{3}}}$.
32. $y^{2}+y=x^{2}$	${\frac {d^{3}y}{dx^{3}}}=-{\frac {24x}{(1+2y)^{5}}}$.
33. $ax^{2}+2hxy+by^{2}=1$.	${\frac {d^{2}y}{dx^{2}}}={\frac {h^{2}-ab}{(hx+by)^{3}}}$.
34. $y^{2}-2xy=a^{2}$.	${\frac {d^{2}y}{dx^{2}}}={\frac {a^{2}}{(y-x)^{3}}};{\frac {d^{3}y}{dx^{3}}}=-{\frac {3a^{2}x}{(y-x)^{5}}}$.
35. $\sec \phi \cos \theta =c$.	${\frac {d^{2}\theta }{d\phi ^{2}}}={\frac {\tan ^{2}\theta -\tan ^{2}\phi }{\tan ^{3}\theta }}$.
36. $\theta \tan(\phi +\theta )$.	${\frac {d^{3}\theta }{d\phi ^{3}}}=-{\frac {2(5+8\theta ^{2}+3\theta ^{4})}{\theta ^{8}}}$.
37. Find the second derivative in the following:
(a) $\log(u+v)=u-v$.	(e) $y^{3}+x^{3}-3axy=0$.
(b) $e^{u}+u=e^{v}+v$.	(f) $y^{2}-2mxy+x^{2}-a=0$.
(c) $s=1+te^{s}$.	(g) $y=\sin(x+y)$.
(d) $e^{s}+st-e=0$.	(h) $e^{x+y}=xy$.

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