Page 284 - DMTH201_Basic Mathematics-1

Unit 10: Logarithmic Differentiation

Notes
x x 2
Example: Differentiate y x e
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
x x 2
y = x e
Apply the natural logarithm to both sides of this equation and use the algebraic properties of
logarithms, getting
x 2
ln y = ln x e x

x 2
= ln x ln e x

e
= x ln x x 2 ln( )
= x ln x x 2 (1)

= x ln x x 2

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 and the chain rule on the right-hand side. Thus,
beginning with

ln y = x ln x x 2

and differentiating, we get

1 1 1/2 1/2
y = x (1/2)x (1/2)x ln x 2x
y x

1 ln x
= 2x
2 x 2 x
(Get a common denominator and combine fractions on the right-hand side.)

1 ln x 2 x
= 2x
2 x 2 x 2 x

1 ln x 4x 1 1/2
=
2 x

1 ln x 4x 3/2
=
2 x

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