Logarithmic Differentiation Calculator

Discover how easy it is to solve functions with tricky logarithmic properties using our logarithmic differentiation calculator.

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Introduction to Logarithmic Differentiation Calculator:

Logarithmic differentiation calculator is the best source of finding the differentiation of a given function using the logarithmic function rules. It is used for the function that has log properties so that it cannot be solved with derivative rules directly.

Logarithmic Differentiation Calculator with Steps

If you want to solve the logarithmic differentiation problem without doing any calculation by hand. Then this tool is made for you as it provides you with a complete solution for your given function without any error quickly and easily.

What is Logarithmic Differentiation?

Logarithmic differentiation is a method that is used in calculus to differentiate functions that are not solved by using standard differentiation rules, like product rule, quotient rule, or chain rule.

This type of function follows logarithmic rules so it can use the natural log to simplify the given function and you can easily solve it with usual differential rules. By using this method you can handle different types of complicated derivative questions without any difficulty.

How to Do Logarithmic Differentiation?

Logarithmic differentiation method can simplify the differentiation process and give you the solution of given functions. Here is a step-by-step explanation of how to calculate the logarithmic differentiation function.

Step 1: Suppose you have a function y that depends on x, which is given as y=f(x).

Step 2: Implement the logarithm on both sides of the given function such as:

$$ ln⁡\; y \;=\; ln\; ⁡(f(x)) $$

Step 3: Now, the differentiation of the equation with respect to x. You can apply any rule of the differential as per the types of function.

Step 4: Solve the given function after applying the derivation rule and you get the solution of the logarithmic problem easily.

If you want to solve logarithmic differentiation problems without doing any calculations, try our derivative of log calculator. It's easy to use and provides accurate solutions in seconds!

Solved Example of Logarithmic Differentiation Function:

The given solved example will give you conceptual clarity about the logarithmic differential function concept.

Example: Determine the derivative of:

$$ y \;=\; (2x^4 + 1)^{tan\; x} $$

Solution:

Take the log on both sides,

$$ ln\; (y) \;=\; ln\; (2x^4 + 1)^{tan\; (x)} $$

$$ ln\; (y) \;=\; tan\; (x) ln\; (2x^4 + 1) $$

Different the log function with respect to x,

$$ \frac{1}{y} \frac{dy}{dx} \;=\; sec^2 (x) ln\; (2x^4 + 1) + \frac{8x^3}{2x^4 + 1} . tan (x) $$

Simplify it to get solution of given function,

$$ \frac{dy}{dx} \;=\; y . \left(sec^2 (x) ln (2x^4 + 1) + \frac{8x^3}{2x^4 + 1} . tan (x) \right) $$

Replace y with original value in solution,

$$ \frac{dy}{dx} \;=\; (2x^4 + 1)^{tan (x)} \left(sec^2 (x)\; ln (2x^4 + 1) + \frac{8x^3}{2x^4 + 1} . tan (x) \right) $$

How to Use the Log Differentiation Calculator?

Log derivative calculator has a simple design that is used for calculating the complex logarithm function derivation without any trouble. Before using this calculator follow some simple steps that you could use for finding the solution of logarithmic derivation. These steps are:

Enter the logarithmic function for differentiation that you would like to differentiate in the input field.
Add the variable of differential for logarithmic function in its input field.
Review your function before clicking on the calculate button to avoid errors in the solution.
Click on the ‘Calculate’ button to get the solution of a given logarithmic function.
Recalculate button will bring you back to the home page for more calculations of the logarithmic function in logarithmic derivative calculator.

Output From Logarithmic Differentiation Calculator With Steps:

Derivative of log calculator gives the result of your given logarithmic function after adding the input function. It may be included as:

Solution Option:

It gives you a solution of your given log function for differentiation.

Possible Step Option:

It provides you with a solution of the logarithmic function in a complete step-by-step calculation process.

Plot Section:

It shows a graph as per your logarithmic differential problem result for better understanding.

Useful Features of Logarithmic Differentiation Calculator:

The Log Differentiation Calculator gives you useful features that you get when you use it to solve the different types of functions, especially the complex logarithmic differentiation functions, and get solutions. The useful features are:

It is a reliable tool that provides you with accurate solutions whenever you use it with no mistakes.
Log derivative calculator is an adaptable tool for everyone because you can use it through electronic devices like laptops, computers tablets, etc.
It is a learning tool that helps students to learn the concept of logarithmic differentiation function easily.
Logarithmic derivative calculator saves time and effort that you put into solving lengthy calculations as it gives you results quickly.
Our tool is free of cost so you do not need to pay anything for the calculation.
Logarithmic differentiation calculator with steps takes you away from any trouble while solving complex logarithmic function derivation by hand.

Related References

What is Logarithmic Differentiation?
Give some examples of logarithmic differentiation:
How to Evaluate logarithmic differentiation?
Explain logarithmic differentiation with the help of examples.

Frequently Ask Questions

When to use logarithmic differentiation

When you use the logarithmic differentiation it depends on the different types of a function. If you have a function that is a product or quotient of other functions then you can use this method for differentiation. When the function has exponent power then you can use the logarithmic function.

For the differential function with complicated algebraic expressions where direct differentiation rules become difficult to solve, the logarithmic differentiation rule provides a simplified method for the differentiation process.

Is the logarithm a differentiable function

Yes, the natural logarithm function ln⁡(x) is differentiable for all x > 0. The derivative of the natural logarithm function ln⁡(x) with respect to x is given by:

$$ \frac{d}{dx} (ln(x)) \;=\; \frac{1}{x},\; for\; x > 0 $$

This derivative can be derived using the definition of the derivative or from the properties of logarithms and the chain rule.

How to do logarithmic differentiation quotient rule

Logarithmic differentiation can be applied to functions that have a quotient rule, it can simplify the process of differentiating functions where the quotient rule alone is difficult to solve. Here’s how you can approach logarithmic differentiation with the quotient rule:

Suppose you have a quotient y = f(x)/g(x), where f(x) and g(x) are functions of x.
implement the logarithm on both sides of the given function such as,

$$ ln⁡\; y \;=\; ln⁡(f(x)g(x)) $$

Then use the properties of logarithms to simplify the quotient function as,

$$ ln⁡\; y \;=\; ln⁡(f(x)) − ln⁡(g(x)) $$

Now take the derivative on both sides of the given function with respect to x.

$$ \frac{1}{y} \frac{dy}{dx} \;=\; \frac{1}{f(x)} \frac{df(x)}{dx} - \frac{1}{g(x)} \frac{dg(x)}{dx} $$

Lasty, solve the above function to get the solution,

$$ \frac{dy}{dx} \;=\; y (\frac{1}{f(x)} \frac{df(x)}{dx} - \frac{1}{g(x)} \frac{dg(x)}{dx}) $$

Why do we use logarithmic differentiation

Logarithmic differentiation is a process in calculus for solving differentiation functions that are complex or difficult to differentiate by direct rules alone (such as product rule, quotient rule, or chain rule).

Logarithmic differentiation simplifies the differentiation of complex functions involving products, quotients, powers, and implicit relationships. It improves accuracy, reduces complexity, and enhances understanding of mathematical concepts.

Why the logarithmic differentiation lnx is 1/x

The reason why the derivative of the natural logarithm function can be understood through calculus principles is given as:

First, take the derivative of a function f(x) with respect to x

$$ f’(x) \;=\; \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} $$

For the natural logarithm function ln⁡(x), this definition leads to:

$$ ln’(x) \;=\; \lim_{h \to 0} \frac{ln(x + h) - ln(x)}{h} $$

After simplifying the limit function,

$$ ln’(x) \;=\; \lim_{h \to 0} \frac{ln(\frac{x+h}{x})}{h} $$

$$ As\; h \to 0,\; \frac{x+h}{x} \to 1 $$

$$ ln’(x) \;=\; \frac{1}{x},\; for\; x > 0 $$

This result of the logarithm function ln⁡(x) is the inverse of the exponential function e^x. The derivative of 1/x is derived from the definition of the derivative of logarithms.

Amanda Jepson

Last Updated February 21, 2024