Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/162

VI	a	$d(x^{n})$	$=nx^{n-1}dx$.
VII		$d\left({\frac {u}{v}}\right)$	$={\frac {vdu-udv}{v^{2}}}$.
VII	a	$d\left({\frac {u}{c}}\right)$	$={\frac {du}{c}}$.
VIII		$d(log_{a}v)$	$=\log _{a}e{\frac {dv}{v}}$.
IX		$d(a^{v})$	$=a^{v}\log adv$.
IX	a	$d(e^{v})=$	$e^{v}dv$.
X		$d(u^{v})$	$=vu^{v-1}du+\log u\cdot u^{v}\cdot dv$.
XI		$d(\sin v)$	$=\cos vdv$.
XII		$d(\cos v)$	$=-\sin vdv$.
XIII		$d(\tan v)$	$=\sec ^{2}vdv$, etc.
XVIII		$d(\arcsin v)$	$={\frac {dv}{\sqrt {1-v^{2}}}}$, etc.

The term "differentiation" also includes the operation of finding differentials.

In finding differentials the easiest way is to find the derivative as usual, and then multiply the result by dx.

Illustrative Example 1. Find the differential of

	$\ y$	$={\frac {x+3}{x^{2}+3}}$
Solution.	$dy=d\left({\frac {x+3}{x^{2}+3}}\right)$	$={\frac {(x^{2}+3)d(x+3)-(x+3)d(x^{2}+3)}{(x^{2}+3)^{2}}}$
	$={\frac {(x^{2}+3)dx-(x+3)2xdx}{(x^{2}+3)^{2}}}$	$={\frac {(3-6x-x^{2})dx}{(x^{2}+3)^{2}}}$, Ans.

Illustrative Example 2. Find dy from

	$b^{2}x^{2}-a^{2}y^{2}$	$a^{2}b^{2}$
Solution.	$2b^{2}xdx-2a^{2}ydy$	$0$.
	∴ $dy$	$={\frac {b^{2}x}{a^{2}y}}dx$. Ans.

Illustrative Example 3. Find dy from

	$\rho ^{2}$	$=a^{2}\cos s\theta$
Solution.	$2\rho d\rho$	$=-a^{2}\sin 2\theta \cdot 2d\theta$.
	∴ $d\rho$	$=-{\frac {a^{2}\sin 2\theta }{\rho }}d\theta$.

Illustrative Example 4. Find $d[\arcsin(3t-4t^{3})]$.

Solution. $d[\arcsin(3t-4t^{3})]={\frac {d(3t-4t^{3})}{\sqrt {1-(3t-4t^{3})^{2}}}}={\frac {3dt}{\sqrt {1-t^{2}}}}$. Ans.