Page 282 - DMTH201_Basic Mathematics-1

Unit 10: Logarithmic Differentiation

Notes
1/x
3x 2 5 6x 2 3x 2 5 ln 3x 2 5
= 2 2 1
x (3x 5)
2
(Combine the powers of (3x + 5).)
(1/x 1)
3x 2 5 6x 2 3x 2 5 ln 3x 2 5
=
x 2

x
Example: Differentiate y (sin )x 3
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 = (sin )x x 3
Apply the natural logarithm to both sides of this equation getting

ln y = ln(sin )x x 3

= x 3 ln(sin )
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 and the chain rule on the right-hand side. Thus,
x
beginning with truein ln y x 3 ln(sin ) and differentiating, we get

1
x
y = x 3 cos x 3x 2 ln(sin )
y sinx
(Get a common denominator and combine fractions on the right-hand side.)

x 3 cosx sinx
x
= 3x 2 ln(sin )
sinx sinx
x
x
x 3 cosx 3x 2 sin ln(sin )
=
sinx
Multiply both sides of this equation by y, getting
x
x
x 3 cosx 3x 2 sin ln(sin )
y = y
sinx
3 x 3 cosx 3x 2 sin ln(sin )
x
x
= (sin )x x
(sin ) 1
x
(Combine the powers of (sin x).)
3
x 1 3 2
x
= (sin )x x cosx 3x sin ln(sin )
x

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