Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/161

EXAMPLES

Find the differential of arc in each of the following curves:

1. $y^{2}=4x$.	Ans.	$ds={\sqrt {\frac {1+x}{x}}}dx.$
2. $y=ax^{2}$.		$ds={\sqrt {1+4a^{2}x^{2}}}dx$.
3. $y=x^{3}$.		$ds={\sqrt {1+9x^{4}}}dx$.
4. $y^{3}=x^{2}$.		$ds={\frac {1}{2}}{\sqrt {4+9y}}dy$.
5. $x^{\frac {2}{3}}+y^{\frac {2}{3}}=a^{\frac {2}{3}}$.		$ds={\sqrt[{3}]{\frac {a}{y}}}dy$.
6. $b^{2}x^{2}+a^{2}y^{2}=a^{2}b^{2}$.		$ds={\sqrt {\frac {a^{2}-e^{2}x^{2}}{a^{2}-x^{2}}}}dx$.
HINT. $e^{2}={\frac {a^{2}-b^{2}}{a^{2}}}$.
7. $e^{y}\cos x=1$.		$ds=\sec \ x\ dx$
8. $\rho =a\cos \theta$.		$ds=a\ d\theta$
9. $\rho ^{2}=a^{2}\cos 2\theta$.		$ds={\sqrt {\sec 2\theta }}d\theta$.
10. $\rho =ae^{\theta \cot \alpha }$.		$ds=\rho \csc \alpha d\theta$.
11. $\rho =a^{\theta }$.		$ds=a^{\theta }{\sqrt {1+\log ^{2}a}}d\theta$.
12. $\rho =a\theta$.		$ds={\frac {1}{a}}{\sqrt {a^{2}+\rho ^{2}}}d\rho$.

13.	(a) $x^{2}-y^{2}=a^{2}$.	(h) $x^{\frac {1}{2}}+y^{\frac {1}{2}}=a^{\frac {1}{2}}$.
	(b) $x^{2}=4ay$.	(i) $y^{2}=ax^{3}$.
	(c) $y=e^{x}+e^{-x}$.	(j) $y=\log x$.
	(d) $xy=a$.	(k) $4x=y^{3}$.
	(e) $y=\log \sec x$.	(l) $\rho =a\sec ^{2}{\frac {\theta }{2}}$.
	(f) $\rho =2a\tan \theta \sin \theta$.	(m) $\rho =1+\sin \theta$.
	(g) $\rho =a\sec ^{3}{\frac {\theta }{3}}$.	(n) $\rho \theta =a$.

92. Formulas for finding the differentials of functions. Since the differential of a function is its derivative multiplied by the differential of the independent variable, it follows at once that the formulas for finding differentials are the same as those for finding derivatives given in § 33, if we multiply each one by dx.

This gives us

I		$d(c)$	$=0$.
II		$d(x)$	$=dx$.
III		$d(u+v-w)$	$=du+dv-dw$.
IV		$d(cv)$	$=cdv$.
V		$d(uv)$	$=udv+vdu$.
VI		$d(v^{n})$	$=nv^{n-1}dv.$