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

11.3 Parametric, implicit and logarithmic differentiation 393
4 (a) 5cosx
5 0,2
(e) 15(t 3 +4t) 14 (3t 2 +4)
1 1 1
8 (a) (b) (c)
(f) −42te −3t2
x x x
4+cosx
1 4 7cosx
(g)
(d) (e) (f)
(4cosx +1) 2 1 +t 3+4t 5+7sinx
(b) 5e x (cosx +sinx)
6 (a) −0.5403 (b) 0 (c) 109.2
(c) 5e sinx cosx
(d) 2 (e) −2.0742
(d) 5e x (cosx +sinx)
7 y=2x+1
11.3 PARAMETRIC,IMPLICITANDLOGARITHMIC
DIFFERENTIATION
11.3.1 Parametricdifferentiation
In some circumstances bothyandxdepend upon a third variable,t. This third variable
is often called a parameter. By eliminatingt, y can be found as a function of x. For
example, ify = (1 +t) 2 andx = 2t then, eliminatingt, we can writey = (1 +x/2) 2 .
Hence,ymay be considered as a function ofx, and so the derivative dy can be found.
dx
However, sometimes the elimination oft is difficult or even impossible. Consider the
exampley=sint+t,x=t 2 +e t .Inthiscase,itisimpossibletoobtainyintermsofx.
The derivative dy can stillbe found usingthe chain rule.
dx
dy
dx = dy
dt × dt
dx = dy / dx
dt dt
Finding dy by thismethod isknown asparametric differentiation.
dx
Example11.7 Giveny = (1 +t) 2 ,x = 2t find dy
dx .
Solution Byeliminatingt, we see
(
y = 1 + x ) 2
=1+x+ x2
2 4
and so
dy
dx = 1 + x 2
Parametricdifferentiationisanalternativemethodoffinding dy
dx whichdoesnotrequire
the elimination oft.
dy
dt =2(1+t)
dx
dt = 2