Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/164

EXAMPLES

Differentiate the following, using differentials:

1. $y=ax^{3}-bx^{2}+cx+d$.	Ans.	$dy=(3ax^{2}-2bx+c)dx$.
2. $y=2x^{\frac {5}{2}}-3x^{\frac {2}{3}}+6x^{-1}+5$.		$dy=(5x^{\frac {3}{2}}-2x^{-{\frac {1}{3}}-6x^{-2}})dx$
3. $y=(a^{2}-x^{2})^{5}$.		$dy=-10x(a^{2}-x^{2})^{4}dx$.
4. $y={\sqrt {1+x^{2}}}$		$dy={\frac {x}{\sqrt {1+x^{2}}}}dx$.
5. $y={\frac {x^{2n}}{(1+x^{2})^{n}}}$.		$dy={\frac {2nx^{2n-1}}{(1+x^{2})^{n+1}}}dx$.
6. $y=\log {\sqrt {1-x^{3}}}$.		$dy={\frac {3x^{2}dx}{2(x^{3}-1)}}$.
7. $y=(e^{x}+e^{-x})^{2}$.		$dy=2(e^{2x}-e^{-2x})dx$.
8. $y=e^{x}\log x$.		$dy=e^{x}\left(\log x+{\frac {1}{x}}\right)dx$.
9. $s=t-{\frac {e^{t}-e^{-t}}{e^{t}+e^{-t}}}$.		$ds=\left({\frac {e^{t}-e^{-t}}{e^{t}+e^{-t}}}\right)^{2}dt$.
10. $\rho =\tan \psi +\sec \psi$		$d\rho ={\frac {1+\sin \psi }{\cos ^{2}\psi }}d\psi$.
11. $r={\frac {1}{3}}\tan ^{3}\theta \tan \theta$.		$dr=\sec ^{4}\theta d\theta$.
12. $f(x)=(\log x)^{3}$.		$f'(x)dx={\frac {3(\log x)^{2}dx}{x}}$.
13. $\psi (t)={\frac {t^{3}}{(1-t^{2}}}^{\frac {3}{2}}$.		$\psi '(t)dt={\frac {3t^{2}dt}{(1-t^{2})^{\frac {5}{2}}}}$.

14. $d\left[{\frac {x\log x}{1-x}}+\log(1-x)\right]={\frac {\log xdx}{(1-x)^{2}}}$.

15. $d[\arctan \log y]={\frac {dy}{y[1+(\log y)^{2}]}}$.

16. $d\left[r\operatorname {arcvers} {\frac {y}{r}}-{\sqrt {2ry-y^{2}}}\right]={\frac {ydy}{\sqrt {2ry-y^{2}}}}$.

17. $d\left[{\frac {\cos \psi }{2\sin ^{2}\psi }}-{\frac {1}{2}}\log \tan {\frac {\psi }{2}}\right]=-{\frac {d\psi }{\sin ^{3}\psi }}$.

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