Logarithmic Differentiation

LOGARITHMIC DIFFERENTIATION

When the given function f(x) is a complicated expression, we take natural logarithm to simplify the function and then differentiate it with respect to x. This technique of differentiation is known as logarithmic differentiation.

It was developed by Bernoulli in 1697.

This method is illustrated in the next few examples

Remember

loge mn = loge m + loge n

loge m/n = loge m - loge n

loge mn= n loge m

WORKED EXAMPLES

Example 1 

Solution 

It is complicated product

Taking log to the base e on both sides, we get

loge y = loge sin2 x + loge tan4 x - loge (x2 + 1)2

= 2 loge sin x + 4 loge tan x − 2 loge (x2 + 1)

Differentiate with respect to x,

Example 2 

Solution 

Given y = xsin x

Since Variable comes in the index to remove it, take loge on both sides

⸫ loge y = sin x loge x

Differentiating with respect to x, we get

Example 3 

Solution 

Taking loge, on both sides we get

loge y = loge sin x loge 10

Differentiating with respect to x, we get

Alternate Method: 

Example 4 

Find the derivative of xsin x + (sin x)x.

Solution 

Let y = xsin x + (sin x)x

Since it is the sum of two terms with variable in the index taking loge as such will not help 

[⸪ we don't have a formula for log(m + n)]

To find these derivatives we take them individually

Let u = xsin x

Taking log on both sides, we get

loge u = sin x loge x

Differentiating with respect to x, we get

Differentiating with respect to x we get,

Example 5 

Solution 

This is the implicit form with variable is x

Given

Taking loge on both sides, we get

Differentiating with respect to x, we get

Example 6 

Solution 

Given

xm yn = (x + y)m+n

Taking loge on both sides, we get

Differentiating with respect to x, we get