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



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.

Find the differential of




 * style="text-align: right;"|$$\ y$$
 * $$= \frac{x + 3}{x^2 + 3}$$
 * Solution.
 * style="text-align: right;"|$$dy = d \left( \frac{x + 3}{x^2 + 3} \right)$$
 * $$= \frac{(x^2 + 3)d(x + 3) - (x + 3)d(x^2 + 3)}{(x^2 + 3)^2}$$
 * style="text-align: right;"|$$= \frac{(x^2 + 3)dx - (x + 3) 2x dx}{(x^2 + 3)^2}$$
 * $$= \frac{(3 - 6x - x^2)dx}{(x^2 + 3)^2}$$, Ans.
 * }
 * style="text-align: right;"|$$= \frac{(x^2 + 3)dx - (x + 3) 2x dx}{(x^2 + 3)^2}$$
 * $$= \frac{(3 - 6x - x^2)dx}{(x^2 + 3)^2}$$, Ans.
 * }
 * }

Find dy from




 * style="text-align: right;"|$$b^2 x^2 - a^2 y^2$$
 * $$a^2 b^2$$
 * Solution.
 * style="text-align: right;"|$$2b^2 x dx - 2a^2 y dy$$
 * $$0$$.
 * style="text-align: right;"|∴ $$dy$$
 * $$= \frac{b^2 x}{a^2 y} dx$$. Ans.
 * }
 * style="text-align: right;"|∴ $$dy$$
 * $$= \frac{b^2 x}{a^2 y} dx$$. Ans.
 * }
 * }

Find dy from




 * style="text-align: right;"|$$\rho^2$$
 * $$= a^2 \cos s\theta$$
 * Solution.
 * style="text-align: right;"|$$2\rho d\rho$$
 * $$= -a^2 \sin 2\theta \cdot 2d\theta$$.
 * style="text-align: right;"|∴ $$d\rho$$
 * $$= -\frac{a^2 \sin 2\theta}{\rho} d\theta$$.
 * }
 * style="text-align: right;"|∴ $$d\rho$$
 * $$= -\frac{a^2 \sin 2\theta}{\rho} d\theta$$.
 * }
 * }

Find $$d[ \arcsin (3t - 4t^3) ]$$.


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