082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

11.3 Parametric, implicit and logarithmic differentiation 399
EXERCISES11.3
1 Find each ofthe following:
(d) x=cos2t y=3t
d d d
(a)
dx (x5 ) (b)
dx (y5 ) (c)
dt (y5 )
(e) x= 3 y=e 2t
t
(f) x=e
d d
(d)
dx (y2 ) (e)
dx (5y) (f) d
t −e −t y=e t +e −t
dx (y) 5 Use logarithmicdifferentiationto find the derivatives
d
ofthe followingfunctions:
(g)
dx (y2 +y 3 )
(a) y =x 4 e x (b) y = 1 x e−x
2 Find eachofthe following:
(c) z =t 3 (1+t) 9 (d) y =e x sinx
d d
(a)
dx (ey ) (b)
dx (siny) (e) y =x 7 sin 4 x
d
(c)
dx (cos2y) (d) d
6 Use logarithmicdifferentiationto find the derivatives
dx (e−3y )
ofthe followingfunctions:
d
(e)
dx (√ d
y) (f)
dx (x+y) (a) z =t 4 (1−t) 6 (2+t) 4
d
(g) (cos(x +y))
dx (h) d
dx (lny)
(b)y= (1 +x2 ) 3 e 7x
(2 +x) 6
(c) x = (1+t) 3 (2+t) 4 (3+t) 5
d
(i)
dx (ln7y)
(d)y= (sin4 t)(2 −t 2 ) 4
3 Find dy
dx given
(1+e t ) 6
(e) y =x 3 e x sinx
(a) 2y 2 −3x 3 =x+y
(b) √ y + √ x=x 2 +y 3
7 Ifx=t+t 2 +t 3 andy=sin2t,find dy whent = 1.
dx
√
(c) 2x+3y=1+e x
8 Givenx=1+t 6 andy=1−t 2 ,find:
(d) y= ex√ 1 +x
(a) the rate ofchange ofxw.r.t.t whent = 2
x 2
(b) the rate ofchange ofyw.r.t.t whent = 1
(e) 2xy 4 =x 3 +3xy 2
(c) the rate ofchange ofyw.r.t.xwhent = 1
(f) sin(x+y)=1+y
(d) the rate ofchange ofxw.r.t.ywhent = 2
(g) ln(x 2 +y 2 ) = 2x −3y
(h) ye 2y =x 2 e x/2
9 Find the equations ofthe tangentsto
4 Find dy
dx ,given
y 2 =x 2 +6y
whenx = 4.
(a) x=t 2 y=1+t 3
(b) x=sint y=e t
10 Given the implicitfunction3x 2 +y 3 =yfindan
(c) x=(1+t) 3 y=1+t 3 expression for dy
dx .
Solutions
1 (a) 5x 4 (b) 5y 4dy
dx
(c) 5y 4dy
dt
(d) 2y dy
dx
(e) 5 dy
dx
(g) (2y +3y 2 ) dy
dx
(f)
dy
dx