Exercises

Differentiate the following by the above rule:

1. $y^2 = 4px$                                                Ans: $\frac{dy}{dx} =\ \frac{2p}{y}$

2. $x^2 + y^2 = r^2$                                                Ans: $\frac{dy}{dx} =\ -\frac{x}{y}$

3. $b^2x^2 + a^2y^2 = a^2b^2$                                                Ans: $\frac{dy}{dx} =\ -\frac{b^2x}{a^2y}$

4. $y^3 - 3y + 2ax = 0$                                                Ans: $\frac{dy}{dx} =\ \frac{2a}{3(1 - y^2)}$

5. $x^{\frac{1}{2}} + y^{\frac{1}{2}} = a^{\frac{1}{2}}$                                                Ans: $\frac{dy}{dx} =\ -\sqrt{\frac{y}{x}}$

6. $x^{\frac{2}{3}} + y^{\frac{2}{3}} = a^{\frac{2}{3}}$                                                Ans: $\frac{dy}{dx} =\ -\sqrt[3]{\frac{y}{x}}$

7. $\left ( \frac{x}{a} \right )^2 + \left ( \frac{y}{b} \right )^{\frac{2}{3}} = 1$                                                Ans: $\frac{dy}{dx} =\ -\frac{3b^{\frac{2}{3}}xy^{\frac{1}{3}}}{a^2}$

8. $y^2 - 2xy + b^2 = 0$                                                Ans: $\frac{dy}{dx} =\ \frac{y}{y - x}$

9. $x^3 + y^3 - 3axy = 0$                                                Ans: $\frac{dy}{dx} =\ \frac{ay - x^2}{y^2 - ax}$

10. $x^y = y^x$                                                Ans: $\frac{dy}{dx} =\ \frac{y^2 - xy\log\, y}{x^2 - xy\log\, x}$

11. $\rho^2 = a^2 \cos 2\theta$                                                Ans: $\frac{d\rho}{d\theta} =\ -\frac{a^2 \sin 2\theta}{\rho}$

12. $\rho^2 \cos\, \theta = a^2 \sin\, 3\theta$                                                Ans: $\frac{d\rho}{d\theta} =\ \frac{3a^2 \cos\, 3\theta + \rho^2 \sin\, \theta}{2\rho \cos\, \theta}$

13. $\cos(uv) = cv$                                                Ans: $\frac{du}{dv} =\ \frac{c + u\sin(uv)}{-v\sin(uv)}$

14. $\theta = \cos(\theta + \phi)$                                                Ans: $\frac{d\theta}{d\phi} =\ -\frac{\sin(\theta + \phi)}{1 + \sin(\theta + \phi)}$

15. Find $\frac{dy}{dx}$ from the following equations:
    $$\begin{array}{lll} (a)\ \ x^2 = ay & (f)\ \ xy + y^2 + 4x = ... ... (j)\ \ y^2 = \sin\, 2x & (o)\ \ e^{x^2} + 2y^3 = 0 \end{array}$$

16. A race track has the form of the circle $x^2 + y^2 = 2500$. The $x$-axis and $y$-axis are east and north respectively, and the unit is $1$ rod^5.14. If a runner starts east at the extreme north point, in what direction will he be going

    (a) when $25\sqrt{2}$ rods east of OY?	Ans. Southeast or southwest.
    (b) when $25\sqrt{2}$ rods north of OX?	Ans. Southeast or northeast.
    (c) when 30 rods west of OY?	Ans. E. $36^o$ 52' 12'' N. or W. $36^o$ 52' 12'' N.
    (d) when 40 rods south of OX?
    (e) when 10 rods east of OY?

17. An automobile course is elliptic in form, the major axis being $6$ miles long and running east and west, while the minor axis is $2$ miles long. If a car starts north at the extreme east point of the course, in what direction will the car be going

    (a) when $2$ miles west of the starting point?

    (b) when $1/2$ mile north of the starting point?