Summary Mathematics for Engineering & the Environment Notes

, Module 3 : Differentiation 1

· First derivative also DEFINITION OF A DERIVATIVE

means first principles
Formally ,
we define the derivative of the

function f(x) at the point to be


lim Af = lim f(x + Ax) =
f(x)
1x- 0 Ax x-0
AR




Two notations are used
:d and f(x) so that




y f(u)
In terms of the function =
,
we write ,




SUMMARY :

· df = f'(x) = lim Af = lim f(x + Au) -
f(u)
AU-O
dx Ar ARxO All

, EXAMPLES




derivative be the definition of first derivatives , find f'(k)
· The
may using when

interpreted as f(x) is

1 . the rate of
change of a) x
the function f(x) f(x) = x
y
=



with respect to x
,
or f(x + Ax) =
(x + Ax)
=
= x+ 2xAx + (Ax)
>




=> f(x + Ax) -
f(x)
Ax
. the
2 slope of the
tangent
at the point (x , y) on the ZuAx + (Ax) = 2x + An
AR
of f(x).
graph y
=



=> d = +'(x) =M(2x + Ax) = 2x




b)
a
- =, futax)
An




Ax

x -
x -
AR
=
Ax(x + Ax)x
- 1

=
x2 + xAx


=> dtim A =I [x= + uxx) 0
= z = -
1x



c) mx + c

f(x) = mx + c
,
f(x + xx) =
m(x + Ax) + c


=> MAx = M

AR

, EXAMPLES




· Rate of change => f' (x)

=> f'(x)


solution




x1 , Y
,