Introduction
The inverse of an exponential function is known as a logarithmic function, and the method to calculate the derivative of this function is called log differentiation. This type of differentiation helps to find the rate of change of inverse or exponential functions. Let’s discuss more logarithmic differentiation and how to apply it step-by-step.
Understanding the Logarithmic Differentiation
Finding the derivative of a logarithmic function is known as logarithmic differentiation. A logarithmic function is the inverse of an exponential function and can be written using a base of 10. It is a method of finding derivatives of complex functions by applying logarithms.
By definition, the derivative of a logarithmic function is defined as:
“The derivative of a function ln y is equal to the reciprocal of y multiplied with the derivative of y without logarithm, is called logarithmic differentiation.”
The logarithmic differentiation follows all derivative rules like the product, power, chain, and quotient rules.
Formula of Logarithmic Differentiation
Any number x that can be written by using some base b is known as a logarithmic function. For a function ln y, the formula of logarithmic differentiation is,
$\frac{d}{dx}[\ln y] = \frac{1}{y}.\frac{dy}{dx}$
Where,
- $\ln y$ is a logarithmic function.
- $\frac{1}{y}$ is the reciprocal of the original function y.
In calculus, the formula to calculate derivative of logarithmic functions can be modified with other derivative rules.
Logarithmic Differentiation and Product Rule
If a function is a product of two functions i.e. one is a logarithmic and the other is any algebraic function, the product rule with the logarithmic differentiation is used to calculate rate of change. For example, the derivative of $F=x\ln x$ can be calculated as;
$\frac{dF}{dx} = x\frac{d}{dx}[\ln x] + ln x\frac{dx}{dx}$
And,
$\frac{dF}{dx} = x(\frac{1}{x}) + \ln x = 1+\ln x$
Logarithmic Differentiation and Power Rule
Since the logarithmic differentiation is used to calculate derivative of a log function. It can be used along with the power rule if the function contains an algebraic function with degree n. The relation between power rule and log differentiation for a function $f(x) = x^m\ln x$ is expressed as;
$f’(x) = \frac{d}{dx}[x^m\ln x]$
By using power rule formula,
$f’(x) = mx^{m-1}(\ln x) + x^m(\frac{1}{x})$
Where, the derivative of ln x with respect to x is 1/x and the derivative of xm with respect to x is $mx^{m-1}$.
Logarithmic Differentiation and Quotient Rule
If a logarithmic function is divided by another function, the logarithmic differentiation along with the quotient rule to find derivative. For a quotient of a function f(x,y) = ln x/x, the relation between partial derivative and quotient rule is,
$\frac{df}{dx} = \frac{(x)\frac{d}{dx}[\ln x] – \ln xd\frac{d}{dx}[x]}{x^2}$
And,
$\frac{df}{dx}= \frac{1 – \ln x}{x^2}$
Where, $\frac{d}{dx}(\ln x) = \frac{1}{x}$.
How do you do logarithmic differentiation step by step?
The implementation of logarithmic derivative is divided into a few steps. These steps assist us to calculate the derivative of a function having more than one independent variable. These steps are:
- Write the expression of the function.
- Identify the logarithmic function.
- Differentiate the function with respect to the variable involved.
- Write the reciprocal of the logarithmic function and multiply it with its derivative without log.
- Simplify if needed.
Let’s understand the following examples by applying logarithmic derivative.
Logarithmic Differentiation Example 1
To calculate the derivative of ln3x, we will use the following steps,
$y =\ln 3x$
Applying the derivative with respect to x.
$\frac{d}{dx}(y) = \frac{d}{dx}(\ln 3x)$
We get,
$\frac{dy}{dx} = \frac{1}{3x} \frac{d}{dx}(3x)$
$\frac{dy}{dx} = \frac{3}{3x} = \frac{1}{x}$
Hence the ln3x derivative is 1/x.
Logarithmic Differentiation Example 2
The derivative of ln x squared can be calculated by using logarithmic differentiation. For this suppose that,
$y = \ln x^2$
Applying the derivative with respect to x.
$\frac{dy}{dx} = \frac{d}{dx}[\ln x^2]$
$\frac{dy}{dx} = \frac{1}{x^2}\frac{d}{dx}(x^2)$
Now by using power rule,
$\frac{dy}{dx} = \frac{1}{x^2} (2x)$
$\frac{dy}{dx} = \frac{2}{x}$
Hence the derivative of ln x squared is 2/x.
Applying logarithmic differentiation formula by using calculator
The derivative of a log function can be also calculated by using a derivative calculator. It is an online tool that follows the log differentiation formula to find derivative. You can find it online by searching for a derivative calculator. For example, to calculate the derivative of ln x, the following steps are used by using this calculator.
- Write the expression of the function in the input box such as, ln x.
- Choose the variable to calculate the rate of change, which will be x in this example.
- Review the input so that there will be no syntax error in the function.
- Now at the last step, click on the calculate button. By using this step, the derivative calculator will provide the derivative of ln x quickly and accurately which will be 1/x
Comparison between Logarithmic and implicit differentiation
The comparison between the logarithmic and implicit differentiation can be easily analysed using the following difference table.
Logarithmic Differentiation | Implicit Differentiation |
The logarithmic differentiation is used to calculate the derivative of a logarithmic function. | The implicit differentiation is used to calculate derivative of an implicit function. |
The derivative of a logarithmic function ln x is defined as; $f’(x) = \frac{1}{x} \frac{d}{dx}(x)$ | There is no specific formula to calculate implicit derivative. |
The logarithmic differentiation can be used along with different derivative formulas. | The implicit differentiation can also be used with derivative rules to find rate of change. |
Conclusion
Logarithmic differentiation is a method to calculate the derivative of a log function. Using logarithmic differentiation, we can calculate the derivative of any complex function. Hence, the formula of log differentiation can be used with the chain rule, product rule, and power rule to calculate the rate of change.