Math Help

Logarithmic Differentiation

In a Nut Shell: Logarithmic differentiation involves taking the natural logarithm of

the function for which the derivative is desired as the first step. It is helpful for more

complicated functions of the types

y = a ^f(x) and y = g(x) ^h(x)

where a is a constant and f(x), g(x), and h(x) are arbitrary functions of x.

The steps in calculating the logarithmic derivative are:

Step 1

Take the natural logarithm on both sides of the equal sign

Step 2

Differentiate implicitly with respect to x

Step 3

Solve the resulting expression for dy/dx .

Suppose y = f(x), then for step (1) ln[ y ] = ln [f(x)]

Then step (2) differentiate implicitly which gives (1/y) dy/dx = d/dx[ln(f(x)]

and finally step (3) dy/dx = y { d/dx[ln(f(x)] }

Example 1 of logarithmic differentiation. Find the derivative of x^x

f(x) = x^x, then taking the natural logarithm yields ln[f(x)] = x ln x

Next take the derivative on both sides of this equation, (implicit differentiation)

f ’(x) / f(x) = ln x + x/x = ln x + 1, then solve for df/dx

So df/dx = f ’(x) = f(x) [ ln x + 1 ] = x^x [ ln x + 1]

Example 2 of logarithmic differentiation. Find the derivative of tanx ^1/x

f(x) = tan x ^1/x, then taking the natural logarithm yields ln[f(x)] = (1/x) ln[tan x]

Next take the derivative on both sides of this equation, (implicit differentiation)

f ’(x) / f(x) = - (1/x²) ln(tan x) + (1/x) sec²x / tan x

So df/dx = tan x ^1/x[ sec²x / x tan x - ln(tan x) / x² ]