Introduction to the Derivative of ln3x
Derivatives have a wide range of applications in almost every field of engineering and science. The derivative of ln(3x) can be calculated by following the rules of differentiation. Or, we can directly find the differentiation of ln3x by applying the first principle of differentiation. In this article, you will learn what the derivative of ln 3x is and how to calculate the derivative of ln3x by using different approaches.
What is the derivative of ln 3x?
The derivative of ln x, also known as the natural logarithmic function, with respect to the variable x is 1/ x. When differentiating ln(3x), it is denoted by d/dx [ln(3x)]. This derivative represents the rate of change of the logarithmic function and is important in calculus. To clarify, ln(3x) can be rewritten as loge 3x, which represents the logarithm of 3x with the base e.
Derivative of ln(3x) formula
The formula for the ln(3x) derivative is equal to the reciprocal of x or 1/x. Mathematically, it is expressed as:
d/dx(ln(3x)) = 1/x
This formula is important in calculus and helps to calculate the rate of change of the natural logarithmic function with respect to the variable x.
How do you prove the differentiation of ln3x?
There are several methods to derive the ln3x derivative. Some of these methods are;
- First Principle
- Implicit Differentiation
- Product Rule
Each method provides a different way to compute the ln(4x) differentiation. By using these methods, we can mathematically prove the formula for finding the ln2x derivative.
Derivative of ln(3x) by first principle
According to the first principle of derivative, the csc 2x derivative is equal to -csc(2x)cot(2x). The derivative of a function by first principle refers to finding a general expression for the slope of a curve by using algebra. It is also known as the delta method. The derivative is a measure of the instantaneous rate of change, which is equal to,
f(x)=lim f(x+h)-f(x) / h
This formula allows us to determine the rate of change of a function at a specific point by using limit definition of derivative.
Proof of ln3x derivative by first principle
To differentiate ln3x by using first principle calculator, we start by replacing f(x) by ln 3x.
f(x)=lim{ln3(x+h)-ln(3x)/h}
By logarithmic properties,
f(x)=lim {ln(x+h/x)/h}
Simplifying,
f(x)=lim {ln(1+h/x)/h}
Suppose t=h / x and h=xt. When h approaches zero, t will also approach zero.
f(x)=lim {ln(1+t)/xt}
And,
f(x)=lim ln (1/xt) ln (1+t)
By logarithmic properties, we can write the above equation as,
f(x)=(1/x) lim ln(1+t)^1/t
Hence by limit formula, we know that,
lim ln(1+t)^1/t =ln e =1
Therefore, the differentiation of ln3x is;
f(x)=1/x
Differential of ln3x using implicit differentiation
Implicit differentiation is a technique used to find the derivative of a function that is defined implicitly by an equation involving two or more variables. We can use this method to prove the differentiation of sin inverse x.
Proof of derivative of ln 3x by implicit differentiation
To differentiate ln 3x we can start by writing it as,
y=ln(3x)
Converting in exponential form,
ey = 3x
Applying derivative on both sides,
d/dx(ey)=d/dx(3x)
ey.dy/dx = 3
Now,
dy/dx=3/ey
Since x=ey
Therefore,
dy/dx=1/x
Use our implicit differential calculator to calculate derivative of implicit functions.
Derivative of ln(3x) using product rule
Another method to calculate the derivative of ln3x is the product rule which is a formula used in calculus to calculate the derivative of the product of two functions. Specifically, the product rule is used when you need to differentiate two functions that are multiplied together. The formula for the product rule calculator is:
d/dx(uv) = u(dv/dx) + (du/dx)v
In this formula, u and v are functions of x, and du/dx and dv/dx are their respective derivatives with respect to x.
Proof of ln3x derivative by product rule
The function ln x can be written as;
f(x)= 1. ln(3x)
Applying derivative with respect to x,
f(x)=(1. ln(3x))
Applying product rule,
f(x)=1.(ln(3x))+ln(3x) (0)
f(x)=1.(1/x)+0
Therefore,
f(x)=1/x
Hence the differential of ln3x is always equal to the reciprocal of x.
How to find the derivative of ln3x with a calculator?
The easiest way to calculate the derivative of ln 3x is by using an online tool. You can use our derivative calculator for this. Here, we provide you a step-by-step way to calculate derivatives by using this differentiation calculator.
- 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 differential of ln3x.
- Now, select the variable by which you want to differentiate ln 3x. Here you have to choose x.
- Select how many times you want to differentiate ln(3x). In this step, you can choose 2 for second, 3 for third derivative and so on.
- Click on the calculate button.
After completing these steps, you will receive the derivative of ln3x proof 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 a logarithmic function?
The derivative of a logarithmic function is the rate of change in a natural log with base e, which is an exponential function. It can be calculated by applying the first principle on the natural log.
What is the derivative of ln(3x)?
The derivative ln3x can be calculated as;
d/dx (ln(3x)) = 1/x