Page 280 - DMTH201_Basic Mathematics-1

Unit 10: Logarithmic Differentiation

Multiply both sides of this equation by y, getting Notes
x
y = y(1 + ln x) = x (1 + ln x)

Example: Differentiate y = x (ex)
Solution:
Because a variable is raised to a variable power in this function, the ordinary rules of
differentiation do not apply ! The function must first be revised before a derivative can be taken.
Begin with
y = x (ex)

Apply the natural logarithm to both sides of this equation getting
x
e
ln y = ln x
Differentiate both sides of this equation. The left-hand side requires the chain rule since y
represents a function of x. Use the product rule on the right-hand side. Thus, beginning with

x
ln y = e ln x
and differentiating, we get

1 x 1 x
y = e e ln x
y x
(Get a common denominator and combine fractions on the right-hand side.)

e x x
= e x ln x
x x

e x xe x lnx
=
x x
e x xe x ln x
=
x
x
(Factor out e in the numerator.)
x
e x (1 x ln )
=
x
Multiply both sides of this equation by y, getting

x
e x (1 x ln )
y = y
x
x
x e x
e (1 x ln )
= x
x 1
(Combine the powers of x.)
x
= x e x 1 e x (1 x ln )

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