Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/55

EXAMPLES

Use the General Rule, p. 291, in differentiating the following functions:

 1. $\scriptstyle {y=3x^{2}.}$Ans. $\scriptstyle {{\frac {dy}{dx}}=6x.}$
 2. $\scriptstyle {y=x^{2}+2.}$Ans. $\scriptstyle {{\frac {dy}{dx}}=2x.}$
 3. $\scriptstyle {y=5-4x.}$Ans. $\scriptstyle {{\frac {dy}{dx}}=-4.}$
 4. $\scriptstyle {s=2t^{2}-4.}$Ans. $\scriptstyle {{\frac {ds}{dt}}=4t.}$
 5. $\scriptstyle {y={\frac {1}{x}}.}$Ans. $\scriptstyle {{\frac {dy}{dx}}=-{\frac {1}{x^{2}}}.}$
 6. $\scriptstyle {y={\frac {x+2}{x}}.}$Ans. $\scriptstyle {{\frac {dy}{dx}}=-{\frac {2}{x^{2}}}.}$
 7. $\scriptstyle {y=x^{3}.}$Ans. $\scriptstyle {{\frac {dy}{dx}}=3x^{2}.}$
 8. $\scriptstyle {y=2x^{2}-3.}$Ans. $\scriptstyle {{\frac {dy}{dx}}=4x.}$
 9. $\scriptstyle {y=1-2x^{3}.}$Ans. $\scriptstyle {{\frac {dy}{dx}}=4x.}$
10. $\scriptstyle {\rho =a\theta ^{2}.}$Ans. $\scriptstyle {{\frac {d\rho }{d\theta }}=2a\theta .}$
11. $\scriptstyle {y={\frac {2}{x^{2}}}.}$Ans. $\scriptstyle {{\frac {dy}{dx}}=-{\frac {4}{x^{3}}}.}$
12. $\scriptstyle {y={\frac {3}{x-1}}.}$Ans. $\scriptstyle {{\frac {dy}{dx}}=-{\frac {3}{(x-1)^{2}}}.}$

13. $\scriptstyle {y=7x^{2}+x.}$
14. $\scriptstyle {s=at^{2}-2bt.}$
15. $\scriptstyle {r=8t+3t^{2}.}$
16. $\scriptstyle {y={\frac {3}{x^{2}}}.}$
17. $\scriptstyle {s=-{\frac {a}{2t+3}}.}$
18. $\scriptstyle {y=bx^{3}-cx.}$
19. $\scriptstyle {\rho =3\theta ^{3}-2\theta ^{2}.}$
20. $\scriptstyle {y={\frac {3}{4}}x^{2}-{\frac {1}{2}}x.}$
21. $\scriptstyle {y={\frac {x^{2}-5}{x}}.}$
22. $\scriptstyle {\rho ={\frac {\theta ^{2}}{1+\theta }}.}$
23. $\scriptstyle {y={\frac {1}{2}}x^{2}+2x.}$
24. $\scriptstyle {z=4x-3x^{2}.}$
25. $\scriptstyle {\rho =3\theta +\theta ^{2}.}$
26. $\scriptstyle {y={\frac {ax+b}{x^{2}}}.}$
27. $\scriptstyle {z={\frac {x^{3}+2}{x}}.}$

28. $\scriptstyle {y=x^{2}-3x+6.}$Ans. $\scriptstyle {y'=2x-3.}$
29. $\scriptstyle {s=2t^{2}+5t-8.}$Ans. $\scriptstyle {s'=4t+5.}$.
30. $\scriptstyle {\rho =5\theta ^{3}-2\theta +6.}$Ans. $\scriptstyle {\rho '=15\theta ^{2}-2.}$
31. $\scriptstyle {y=ax^{2}+bx+c.}$Ans. $\scriptstyle {y'=2ax+b.}$

32. Applications of the derivative to Geometry. We shall now consider a theorem which is fundamental in all applications of the Differential Calculus to Geometry. Let

(A)	$\scriptstyle {y=f(x)}$

be the equation of a curve $\scriptstyle {AB}$.

Now differentiate (A) by the General Rule and interpret each step geometrically.

First Step.	$\scriptstyle {y+\Delta y}$	$\scriptstyle {=f(x+\Delta x)}$	$\scriptstyle {=NQ}$
Second Step.	$\scriptstyle {y+\Delta y}$	$\scriptstyle {=f(x+\Delta x)}$	$\scriptstyle {=NQ}$
	$\scriptstyle {y}$$\scriptstyle {+\Delta y}$	$\scriptstyle {=f(x)}$	$\scriptstyle {=MP=NR}$
	$\scriptstyle {\Delta y}$	$\scriptstyle {=f(x+\Delta x)-f(x)}$	$\scriptstyle {=RQ.}$

Third Step.	$\scriptstyle {\frac {\Delta y}{\Delta x}}$	$\scriptstyle {={\frac {f(x+\Delta x)-f(x)}{\Delta x}}={\frac {RQ}{MN}}={\frac {RQ}{PR}}}$
		$\scriptstyle {=\tan RPQ=\tan \phi }$
		=slope of secant line $\scriptstyle {PQ.}$