Examples

Differentiate the following, using differentials:

1. $y = ax^3 - bx^2 + cx + d$.

   Ans. $dy = (3ax^2 - 2bx + c)dx$.

2. $y = 2x^{\frac{5}{2}} - 3x^{\frac{2}{3}} + 6x^{-1} + 5$.

   Ans. $dy = (5x^{\frac{3}{2}} - 2x^{-\frac{1}{3}} - 6x^{-2})dx$.

3. $y = (a^2 - x^2)^5$.

   Ans. $dy = - 10x(a^2 - x^2)^4dx$.

4. $y = \sqrt{1 + x^2}$.

   Ans. $dy = \frac{x}{\sqrt{1 + x^2}} dx$.

5. $y = \frac{x^{2n}}{(1 + x^2)^n}$.

   Ans. $dy = \frac{2nx^{2n - 1}}{(1 + x^2)^{n + 1}} dx$.

6. $y = \log \sqrt{1 - x^3}$.

   Ans. $dy = \frac{3x^2 dx}{2(x^3 - 1)}$.

7. $y = (e^x + e^{-x})^2$.

   Ans. $dy = 2(e^{2x} - e^{-2x})dx$.

8. $y = e^x\log\, x$.

   Ans. $dy = e^x \left ( \log x + \frac{1}{x} \right ) dx$.

9. $s = t - \frac{e^t - e^{-t}}{e^t + e^{-t}}$.

   Ans. $ds = \left ( \frac{e^t - e^{-t}}{e^t + e^{-t}} \right )^2 dt$.

10. $\rho = \tan\psi + \sec\psi$.

    Ans. $d\rho = \frac{1 + \sin \psi}{\cos^2 \psi} d\psi$.

11. $r = \frac{1}{3} \tan^3 \theta \tan \theta$.

    Ans. $dr = \sec^4\theta d\theta$.

12. $f(x) = (\log\, x)^3$.

    Ans. $f'(x) dx = \frac{3(\log x)^2 dx}{x}$.

13. $\psi(t) = \frac{t^3}{(1 - t^2)^{\frac{3}{2}}}$.

    Ans. $\psi'(t) dt = \frac{3 t^2 dt}{(1 - t^2)^{\frac{5}{2}}}$.

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

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

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

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