Derivative of ln(u): Formula, Proof, Examples, Solution

Introduction to the derivative of ln

Derivatives have a wide range of applications in almost every field of engineering and science. The ln u derivative can be calculated by following the derivative rules. Or, we can directly find the ln derivative by applying the first principle of differentiation. In this article, you will learn what the derivative ln u is and how to calculate the derivative of ln(u) by using different approaches.

What is the derivative of ln(u)?

The ln(u) derivative is equal to 1/u and is represented by d/dx(ln x. The derivative of ln u with respect to u is a crucial concept in calculus. This derivative describes the rate of change of the natural logarithmic function ln(x), which is used extensively in various mathematical and scientific fields.

Additionally, the expression e raised to the power of ln u is equal to u, which highlights the significance of the derivative in understanding the logarithmic properties of u. Therefore, understanding the derivative of ln(u) is essential for mastering calculus and related subjects. Mathematically the relation between e and ln u is expressed as;

$e^{\ln u} = u$

It represents the logarithm of u with base e Similarly, the derivative of ln x is:

.$e^{\ln x} = x$

Derivative of ln formula

The derivative of natural logarithm, ln u, is calculated using the formula,

$\frac{d}{du} (\ln u) = \frac{1}{u}$

This formula can be proven using the limit definition of a derivative. The ln(u) derivative is an essential concept in calculus and finds application in various mathematical and scientific fields. By understanding the derivative of ln u, you can solve derivative problems related to optimization, rates of change, and exponential growth.

How do you prove the ln u derivative formula?

There are different ways or methods to derive derivatives of ln u. These methods allow you to find the rate of change or slope of the ln u function at any point, which is useful in many applications. We can prove the ln u derivative by using;

First Principle
Implicit Differentiation
Product Rule

Derivative of ln(u) by first principle

The derivative first principle tells that the differentiation of ln u is equal to the 1/u. The derivative of a function by first principle refers to finding the slope of a curve by using algebra. It is also known as the delta method. Mathematically, the first principle of derivative formula is represented as:

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

You can also use our derivative by definition calculator to easily calculate the derivative of log functions.

Proof of derivative of ln u by first principle

To prove the derivative of ln x by using first principle, we start with replacing f(x) by ln u and to differentiate ln3x, replace f(x) by ln(3x).

$f'(u) \;=\; \lim_{h \to 0}\frac{ln (u+h) -ln u}{h}$

By logarithmic properties,

$f'(u) \;=\; \lim_{h \to 0}\frac{\frac{ln(u+h)}{u}}{h}$

Simplifying,

$f'(u) \;=\; \lim_{h \to 0}\frac{ln(\frac{1+h}{u})}{h}$

Suppose t = h / u and h=ut. When h approaches zero, t will also approach zero.

$f'(x) \;=\; \lim_{t to 0} \frac{ln (1+t)}{ut}$

And,

$f'(u) \;=\; \lim_{t \to 0}\frac{1}{ut} ln (1+t)$

By logarithmic properties, we can write the above equation as,

$f'(x) \;=\; \frac{1}{u}\lim_{t to 0}\ln (1+t)^{\frac{1}{t}}$

Hence by limit formula, we know that,

$\lim_{t \to 0}\ln (1+t)^{\frac{1}{t}} \;=\; \ln e \;=\; 1$

Therefore, the ln u derivative formula is;

$f'(x) \;=\; (\frac{1}{u})$

This means that the slope of the curved line at any point on the graph of ln u is equal to the reciprocal of u.

d/dx ln u using implicit differentiation

Since in implicit differentiation, we differentiate a function with two variables. Here we will prove the ln derivative by implicit differentiation.

Proof of ln u derivative by implicit differentiation

We can easily differentiate ln u by using product rules. Now to prove the derivative of natural log, we can write it as,

$y \;=\; \ln u$

Converting in exponential form,

$e^y \;=\;u$

Applying derivative on both sides,

$\frac{d}{du}(e^y) \;=\; \frac{d}{du}(u)$

$e^y.\frac{dy}{du} \;=\; 1$

Now,

$\frac{dy}{du} \;=\; \frac{1}{e^y}$

Since

$e^y \;=\; u$

Therefore, the derivative of ln(u) is,

$\frac{dy}{du} \;=\; \frac{1}{u}$

Derivative of ln u using product rule

The product rule in derivatives is used when we have to calculate derivatives of two functions at a time. The product rule for two functions says that;

$\frac{d}{du}(f(u)g(u)) \;=\; f’(u)g(u) \;+\; g’(u)f(u)$

Since the function ln u can be written as a product of two functions, therefore we can use it to prove its derivative.

ln u derivative formula proof by product rule

To find the logarithmic differentiation of the function ln(u), the function ln u can be written as;

$f(u) \;=\; 1.\ln u$

Applying derivative with respect to u,

$f'(u) \;=\;(1.\ln u )'$

Applying product rule,

$f'(u) \;=\;1.u +\ln u. (1)$

$f'(u)=1.\frac{1}{u} + 0$

Therefore,

$f'(u) \;=\; \frac{1}{u}$

Hence the d/dx ln u is always equal to the reciprocal of u. You can also try a product rule calculator with steps to differentiate ln u easily.

How to find the derivative of ln u with a calculator?

The easiest way to calculate the ln derivative is by using an online derivative finder. You can use our derivative calculator for this. Here, we provide you a step-by-step way to calculate derivatives by using this tool.

Write the function as ln x in the enter function box. In this step, you need to provide input value as a function as you have to calculate the derivative of the function ln x.
Now, select the variable by which you want to differentiate ln u. Here you have to choose u.
Select how many times you want to calculate derivatives of ln. In this step, you can choose 2 for second, 3 for third order derivative and so on.
Click on the calculate button. After this step, you will get the d/dx ln u within a few seconds.

After completing these steps, you will receive the derivative of ln x within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions.

Frequently Asked Questions

What is the derivative of ln u?

The ln u derivative is always equal to 1/u. Because when we differentiate a function with natural log, it always results in the reciprocal of the function. The derivative ln u is written as;