(10) Rate of change of P when t varies Rate of change of P when D varies = − D t {\displaystyle {\dfrac {\text{Rate of change of P when t varies}}{\text{Rate of change of P when D varies}}}=-{\dfrac {D}{t}}} .
(11) 2 π {\displaystyle 2\pi } , 2 π r {\displaystyle 2\pi r} , π l {\displaystyle \pi l} , 2 3 π r h {\displaystyle {\tfrac {2}{3}}\pi rh} , 8 π r {\displaystyle 8\pi r} , 4 π r 2 {\displaystyle 4\pi r^{2}} .
(12) d D d T = 0.000012 l t π {\displaystyle {\dfrac {dD}{dT}}={\dfrac {0.000012l_{t}}{\pi }}} .
Exercises III. (p. 46.)
(1) (a) 1 + x + x 2 2 + x 3 6 + x 4 24 + … {\displaystyle 1+x+{\dfrac {x^{2}}{2}}+{\dfrac {x^{3}}{6}}+{\dfrac {x^{4}}{24}}+\ldots } .
(2) d w d t = a − b t {\displaystyle {\dfrac {dw}{dt}}=a-bt} .
(3) d y d x = 2 x {\displaystyle {\dfrac {dy}{dx}}=2x} .
(4) 14110 x 4 − 65404 x 3 − 2244 x 2 + 8192 x + 1379 {\displaystyle 14110x^{4}-65404x^{3}-2244x^{2}+8192x+1379} .
(5) d x d y = 2 y + 8 {\displaystyle {\dfrac {dx}{dy}}=2y+8} .
(6) 185.9022654 x 2 + 154.36334 {\displaystyle 185.9022654x^{2}+154.36334} .
(7) − 5 ( 3 x + 2 ) 2 {\displaystyle {\dfrac {-5}{(3x+2)^{2}}}} .
(8) 6 x 4 + 6 x 3 + 9 x 2 ( 1 + x + 2 x 2 ) 2 {\displaystyle {\dfrac {6x^{4}+6x^{3}+9x^{2}}{(1+x+2x^{2})^{2}}}} .
(9) a d − b c ( c x + d ) 2 {\displaystyle {\dfrac {ad-bc}{(cx+d)^{2}}}} .
(10) a n x − n − 1 + b n x n − 1 + 2 n x − 1 ( x − n + b ) 2 {\displaystyle {\dfrac {anx^{-n-1}+bnx^{n-1}+2nx^{-1}}{(x^{-n}+b)^{2}}}} .
(11) b + 2 c t {\displaystyle b+2ct} .
(12) R 0 ( a + 2 b t ) {\displaystyle R_{0}(a+2bt)} , R 0 ( a + b 2 t ) {\displaystyle R_{0}\left(a+{\dfrac {b}{2{\sqrt {t}}}}\right)} , − R 0 ( a + 2 b t ) ( 1 + a t + b t 2 ) 2 {\displaystyle -{\dfrac {R_{0}(a+2bt)}{(1+at+bt^{2})^{2}}}} or R 2 ( a + 2 b t ) R 0 {\displaystyle {\dfrac {R^{2}(a+2bt)}{R_{0}}}} .
(13) 1.4340 ( 0.000014 t − 0.001024 ) {\displaystyle 1.4340(0.000014t-0.001024)} , − 0.00117 {\displaystyle -0.00117} , − 0.00107 {\displaystyle -0.00107} , − 0.00097 {\displaystyle -0.00097} .
(14) d E d l = b + k i {\displaystyle {\dfrac {dE}{dl}}=b+{\dfrac {k}{i}}} , d E d i = − c + k l i 2 {\displaystyle {\dfrac {dE}{di}}=-{\dfrac {c+kl}{i^{2}}}} .
Exercises IV. (p. 51.)
(1) 17 + 24 x {\displaystyle 17+24x} ; 24 {\displaystyle 24} .
(2) x 2 + 2 a x − a ( x + a ) 2 {\displaystyle {\dfrac {x^{2}+2ax-a}{(x+a)^{2}}}} ; 2 a ( a + 1 ) ( x + a ) 3 {\displaystyle {\dfrac {2a(a+1)}{(x+a)^{3}}}}
(3) 1 + x + x 2 1 × 2 + x 3 1 × 2 × 3 {\displaystyle 1+x+{\dfrac {x^{2}}{1\times 2}}+{\dfrac {x^{3}}{1\times 2\times 3}}} .
(4) (Exercises III.):
(1) (a) d 2 y d x 2 = d 3 y d x 3 = 1 + x + 1 2 x 2 + 1 6 x 3 + … {\displaystyle {\dfrac {d^{2}y}{dx^{2}}}={\dfrac {d^{3}y}{dx^{3}}}=1+x+{\frac {1}{2}}x^{2}+{\frac {1}{6}}x^{3}+\ldots }