Applications of Log Differentiation

Log differentiation found its application while solving various differentiation problems. Various types of problems where Log Differentiation is used are discussed below,

Product of Functions (Product Rule)

The differentiation of any function which is a product of two functions can easily be calculated using logarithmic differentiation.

Suppose we have to find differentiation of f(x) where, f(x) = g(x) × h(x) then by using concept of logarithmic differentiation,

f(x) = g(x) × h(x)

Taking log on both sides, 

log f(x) = log (g(x) × h(x))

⇒ log f(x) = log g(x) + log h(x)        [Using property log (XY) = log (X) + log (Y)]

Differentiating both sides with respect to x,

d/dx [log f(x)] = d/dx [log g(x)] + d/dx [log h(x)]

⇒ f'(x)/f(x) = g'(x)/g(x) + h'(x)/h(x)

⇒ f'(x) = f(x) [g'(x)/g(x) + h'(x)/h(x)]

⇒ f'(x) = f(x) [(h(x)×g'(x) + g(x)×h'(x))/ (g(x)×h(x))]

⇒ f'(x) = g(x)×h(x) [h(x)×g'(x) + g(x)×h'(x)] / g(x)×h(x)

∴ f'(x) = h(x)×g'(x) + g(x)×h'(x)

The result obtained above is the “Leibniz rule” and is commonly known as the “Product rule“.

Division of Functions (Quotient Rule)

The differentiation of any function which is in the form of a division of two functions can easily be calculated using logarithmic differentiation.

Suppose one has to find the differentiation of f(x) where f(x) = g(x) / h(x) by using the concept of logarithmic. differentiation,

f(x) = g(x)/h(x)

Taking log on both sides,

log f(x) = log [g(x)/h(x)]

⇒ log f(x) = log g(x) – log h(x)                       [Using property log (X/Y) = log (X) – log (Y)]

Differentiating both sides with respect to x,

d/dx [log f(x)] = d/dx [log g(x)] – d/dx [log h(x)]

⇒ f'(x)/f(x) = g'(x)/g(x) – h'(x)/h(x)

⇒ f'(x) = f(x)[g'(x)/g(x) – h'(x)/h(x)]

⇒ f'(x) = f(x) [(g'(x)×h(x) – g(x)×h'(x))/(g(x)×h(x))]

⇒ f'(x) = g(x)/h(x) [g'(x)×h(x) – g(x)×h'(x)]/g(x)×h(x)

∴ f'(x) = [g'(x)×h(x) – g(x)×h'(x)] / h²(x)

The result obtained above is commonly known as the “Quotient rule“.

Exponential Functions

The differentiation of any function which is in exponential form can easily be calculated using logarithmic differentiation.

Suppose we have to find differentiation of f(x) where, f(x) = g(x)^h(x) then by using concept of logarithmic differentiation,

f(x) = g(x)^h(x)

Taking log on both sides

log f(x) = log [g(x)^h(x)]

⇒ log f(x) = h(x) × log (g(x))              [Using property log (X^Y) = Ylog (X)]

Differentiating both sides with respect to x,

d/dx [log f(x)] = d/dx [h(x) × log g(x)]

⇒ f'(x)/f(x) = h(x) × d/dx log g(x) + log g(x) × d/dx h(x)    [Using uv rule of differentiation]

⇒ f'(x)/f(x) = h(x) × g'(x)/g(x) + log g(x) × h'(x)

⇒ f'(x)/f(x) = [h(x) × g'(x) + g(x) × h'(x) × log g(x)] / g(x)

⇒ f'(x) = f(x) [h(x)×g'(x) + g(x)×h'(x)×log g(x)] / g(x)

⇒ f'(x) = g(x)^h(x) [h(x)×g'(x) + g(x)×h'(x)×log g(x)] / g(x)

∴ f'(x) = g(x)^h(x)-1 [h(x)×g'(x) + g(x)×h'(x)×log g(x)]

Logarithmic Differentiation helps to find the derivatives of complicated functions, using the concept of logarithms. Sometimes finding the differentiation of the function is very tough but differentiating the logarithm of the same function is very easy, then in such cases, the logarithmic differentiation formula is used. 

In calculus, the differentiation of some complex functions is found first by taking a log and then finding the logarithmic derivative of that function.

In this article, we will learn about Logarithmic Differentiation in detail.