Differentiation

Derivative of

Logarithmic Function

Differentiation

We take the natural logarithm of both sides of the equation y = ( f (x))g(x), to btain 

ln y = ln ( f (x))g(x) = g(x) ln f (x)

Then we differentiate implicitly both sides of the resulting equation ln y = g(x) ln f (x) with respect to x.

Logarithmic Differentiation The Function y = ( f (x)) g (x)

Differentiation

PROPERTIES OF THE NATURAL LOGARITHM

Differentiation

Example 1:

Solution

Differentiation

Example 2: Differentiate y = xx 

Solution Apply the natural logarithm to both sides of this equation getting

Differentiate both sides of this equation. 

Multiply both sides of this equation by y, getting

Differentiation

Example 3: Differentiate

Solution

Differentiation

Example 4: Differentiate Solution

Differentiation

Example 5: Differentiate

Solution

Differentiation

Derivative of Logarithmic Function

ASSESSMENT

Differentiation

Using logarithmic differentiation, differentiate:

a:

b:

c:

d:

Differentiation

e:

f:

Differentiation

a:

b:

c:

d:

Solutions:

Differentiation

e:

f: