Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/125

EXAMPLES

Differentiate the following:

1. $y=4x^{3}-6x^{2}+4x+7$.	${\frac {d^{2}y}{dx^{2}}}=12(2x-1)$.
2. $f(x)={\frac {x^{3}}{1-x}}$.	$f^{iv}(x)={\frac {4!}{(1-x)^{5}}}$.
3. $f(y)=y^{6}$.	$f^{vi}(y)=6!$.
4. $y=x^{3}\log x$.	${\frac {d^{4}y}{dx^{4}}}={\frac {6}{x}}$.
5. $y={\frac {c}{x^{n}}}$.	$y''={\frac {n(n+1)c}{x^{n+2}}}$.
6. $y=(x-3)e^{2x}+4xe^{x}+x$.	$y''=4e^{x}[(x-2)e^{x}+x+2]$.
7. $y={\frac {a}{2}}(e^{\frac {x}{a}}+e^{-{\frac {x}{a}}})$.	$y''={\frac {1}{2a}}(e^{\frac {x}{a}}+e^{-{\frac {x}{a}}})={\frac {y}{a^{2}}}$.
8. $f(x)=ax^{2}+bx+c$.	$f'''(x)=0$.
9. $f(x)=\log(x+1)$.	$f^{iv}(x)=-{\frac {6}{(x+1)^{4}}}$.
10. $f(x)=\log(e^{x}+e^{-x})$.	$f'''(x)=-{\frac {8(e^{x}-e^{-x})}{(e^{x}-e^{-x})^{3}}}$.
11. $r=\sin a\theta$.	${\frac {d^{4}r}{d\theta ^{4}}}=a^{4}\sin a\theta =a^{4}r$.
12. $r=\tan \phi$.	${\frac {d^{3}r}{d\phi ^{3}}}=6\sec ^{6}\phi -4\sec ^{2}\phi$.
13. $r=\log \sin \phi$	$r'''=2\cot \phi \csc ^{2}\phi$.
14. $f(t)=e^{-t}\cos t$.	$f^{iv}(t)=-4e^{-t}\cos t=-4f(t)$.
15. $f(\theta )={\sqrt {\sec 2\theta }}$.	$f''(\theta )=3[f(\theta )]^{5}-f(\theta )$.
16. $p=(q^{2}+a^{2})\arctan {\frac {q}{a}}$.	${\frac {d^{3}p}{dq^{3}}}={\frac {4a^{3}}{(a^{2}+q^{2})^{2}}}$.
17. $y=a^{x}$.	${\frac {d^{n}y}{dx^{n}}}=(\log a)^{n}a^{x}$.
18. $y=\log(1+x)$.	${\frac {d^{n}y}{dx^{n}}}=(-1)^{n-1}{\frac {(n-1)!}{(1+x)^{n}}}$.
19. $y=\cos ax$	${\frac {d^{n}y}{dx^{n}}}=a^{n}\cos \left(ax+{\frac {n\pi }{2}}\right)$.
20. $y=x^{n-1}\log x$	${\frac {d^{n}y}{dx^{n}}}={\frac {(n-1)!}{x}}$.
[n = a positive integer]
21. $y={\frac {1-x}{1+x}}$.	${\frac {d^{n}y}{dx^{n}}}=2(-1)^{n})={\frac {n!}{(1+x)^{n+1}}}$
HINT. Reduce fraction to form $-1+{\frac {2}{1+x}}$ before differentiating.
22. If $y=e^{x}\sin x$, prove that ${\frac {d^{2}y}{dx^{2}}}-2{\frac {dy}{dx}}+2y=0$.
23. If $y=a\cos(\log x)+b\sin(\log x)$, prove that $x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+y=0$.