Page 9 - PRE-U STPM MATHEMATICS (T) TERM 2

Mathematics Term 2 STPM Chapter 2 Differentiation
But y → 0 when x → 0,
dy 1
=
dx lim 1 2
x
x → 0 y
dy 1
=
dx dx
dy

When ln x = y, From the definition of logarithm i.e. if log y = x, then y = a .
x
x = e y a
Differentiate both sides w.r.t. y,
2 dx d
y
y
= (e ) = e = x
dy dy
dy 1 1
= =
dx dx x
dy
d 1
Hence, (ln x) =
dx x

Derivative of a x

Let y = a x
Taking log of both sides to base e,
ln y = ln a x
= x ln a

Differentiate both sides w.r.t. y,
d dx
(ln y) = ln a
dy dy
1 dx
y = dy ln a

dx 1
=
dy y ln a
dy
dx = y ln a
= a ln a
x
Hence, d (a ) = a ln a
x
x
dx
Example 4

Differentiate each of the following with respect to x.
x
(a) 3 (b) 10 x
Solution: (a) Let y = 3 x (b) y = 10 x
dy dy
x
Hence, = 3 ln 3 Hence, = 10 ln 10
x
dx dx

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02 STPM Math(T) T2.indd 22 02/11/2018 12:43 PM