Logarithmic Differentiation Formula
For a function, y = f(x)g(x), differentiation is given by the following formula:
[Tex]\bold{\frac{dy}{dx} = y\left[g(x)\cdot \frac{f'(x)}{f(x)} + log(f(x))g'(x)\right]} [/Tex]
Logarithmic formulas are very useful in solving logarithmic differentiation. Some of the important logarithmic properties used are,
- log XY = log X + log Y
- log X/Y = log X – log Y
- log XY = Y log X
- logY X = (log X) / (log Y)
Note: Logarithmic differentiation rules are only valid for the positive functions only because logarithm of negative function is undefined.
Derivation of Logarithmic Differentiation Formula
Let us consider a function y = f(x)g(x),
and take the natural logarithm of this function to differentiate it,
ln y = ln (f(x)g(x))
⇒ ln (y) = g(x) ln (f(x))
Differentiate the above equation,
[Tex]\frac{d [\ln y]}{dx} = \frac{d}{dx}[g(x) \cdot \ln f(x)] [/Tex]
[Tex]\Rightarrow \frac{1}{y} \cdot \frac{dy}{dx} = g'(x) \ln(f(x)) + g(x) \cdot \frac{d}{dx}(\ln(f(x))) [/Tex]
[Tex]\Rightarrow \frac{dy}{dx} = y \left(g'(x) \ln(f(x)) + g(x) \cdot \frac{d}{dx}(\ln(f(x)))\right) [/Tex]
[Tex]\Rightarrow \frac{dy}{dx} = y \left[g'(x) \ln(f(x)) + g(x) \cdot \frac{f'(x)}{f(x)}\right] [/Tex]
Which is the required formula.
Logarithmic Differentiation
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.
Table of Content
- What is Logarithmic Differentiation?
- Logarithmic Differentiation Formula
- Derivation of Logarithmic Differentiation Formula
- Applications of Log Differentiation
- Product of Functions (Product Rule)
- Division of Functions (Quotient Rule)
- Exponential Functions
- Method to Solve Logarithmic Functions
- Solved Examples on Logarithmic Differentiation