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

366 Chapter 10 Differentiation
then
Usingthe previous example, if
y(x) =2x 2 +3x
dy
dx =4x+3
dy
isoften abbreviated toy′
dx
y ′ is pronounced ‘y dash’ or ‘y prime’. To stress thatyis the dependent variable andx
the independent variable we often talk of ‘the rate of change ofywith respect tox’, or,
more compactly, ‘the rate of change of y w.r.t. x’. The process of finding y ′ from y is
calleddifferentiation.Thisshrinkingintervalmethodoffindingthederivativeiscalled
differentiation from first principles. We know that the derivative dy is the gradient
dx
of the tangent to the function at a point. It is also the rate of change of the function. In
manyexamples,theindependentvariableist andweneedtofindtherateofchangeofy
withrespecttot;thatis,find dy
dt .Thisisalsooftenwrittenasy′ althoughẏ,pronounced
‘ydot’,isalsocommon.Thereadershouldbeawareofbothnotations.Finally,y ′ isused
to denote the derivative ofywhatever the independent variable may be. So dy
dz , dy
dr and
dy
dw could all be represented byy′ .
Example10.7 Findthe gradientofthe tangent toy =x 2 atA(1,1),B(−1,1) andC(2,4).
Solution We havey =x 2 and so
y(x + δx) = (x + δx) 2
and hence
Then
=x 2 +2xδx + (δx) 2
y(x + δx) −y(x) =x 2 +2xδx + (δx) 2 −x 2
dy
dx
= 2xδx + (δx) 2
= gradient of a tangent tocurve
( ) y(x + δx) −y(x)
= lim
δx→0 δx
)
= lim
δx→0
( (x + δx) 2 −x 2
δx
= lim
δx→0
(2x + δx) =2x
( ) 2xδx + (δx)
2
= lim
δx→0 δx