Introduction to the derivative of ln8x

Derivatives have a wide range of applications in almost every field of engineering and science. The derivative of ln8 can be calculated by following the rules of differentiation. Or, we can directly find the derivative of ln(8x) by applying the first principle of differentiation. In this article, you will learn what the ln(8x) derivative is and how to calculate the differentiation of ln(8x) by using different approaches. 

What is the derivative of ln(8x)?

The derivative of ln8x with respect to x is equal to 1/x. This can be expressed as d/dx ln(8x). It represents the rate of change of the natural logarithmic function ln(x) and is written as:

$\ln 8x=\log_e 8x$

The expression loge(8x) represents the logarithm of 8x with base e. This is a useful way to express the natural logarithm of 8x, because it helps us to easily calculate its derivative and understand its relationship with other mathematical functions.

Ln8x derivative formula

The differentiation of ln(8x) is equal to 1/x. It can be calculated the same as the derivative of ln(x). Applying this formula to ln(8x), you can simplify it to:

$\frac{d}{dx}(\ln 8x) =\frac{1}{x}$

This means that the derivative of ln(8x) is equal to the reciprocal of x. It means that the rate of change of the ln(8x) with respect to x is inversely proportional to x. This property of logarithmic functions can be useful in many applications, such as modeling population growth or analyzing financial data.

How do you prove the derivative of ln(8x)?

 There are numerous ways to derive derivatives of ln(8x). Therefore, we can prove the derivative of ln 8x by using;

1. First Principle
2. Implicit Differentiation
3. Product Rule

Derivative of ln8 by first principle

The first principle of differentiation tells us that to find the derivative of ln(8x), we can use algebra to find a general expression for the slope of the curve at any point. This method is also called the delta method. The derivative measures the instantaneous rate of change of a function with respect to its independent variable. For ln(8x), the derivative is equal to the reciprocal of x, which represents the slope of the tangent line to the curve at any given point. Mathematically,

f'(x)=lim f(x+h)-f(x)/h

This formula can be applied in a variety of calculus problems, including optimization and related rates problems. Understanding the first principle of differentiation is important because it provides a fundamental concept for calculating the derivative of any function.

Proof of derivative ln(8x) by first principle

To prove the ln(8x) derivative by using derivative definition calculator, replace f(x) by ln 8x. 

f’(x)=lim{ln8(x+h)-ln(8x)/h}

By logarithmic properties,

f’(x)=lim {ln(x+h/x)/h}

Simplifying,

f’(x)=lim {ln(1+h/x)/h}

Suppose t=hx 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)/t

Hence by limit formula, we know that,

lim ln(1+t)/t =ln e =1

Therefore, the derivative ln(8x) is;

f’(x)=1/x

Also learn to calculate the derivative of ln7x by using first principle of derivative.

Derivative of ln 8x 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 ln(8x).

Proof of derivative of ln(8x) by implicit differentiation 

 To prove the derivative of natural log, we can write it as,

y=ln(8x)

Converting in exponential form, 

ey = 8x

Applying derivative on both sides,

d/dx(ey)=d/dx(8x)

ey.dy/dx = 8

Now,

dy/dx=8/ey

Since x=ey

Therefore, 

dy/dx=1/x

Another easy way to calculate rate of change of the function ln8x is an implicit derivative calculator online. It provides you a step-by-step way of solving derivatives.

Derivative of ln(8x) using product rule

The product rule is a rule in calculus that is used to find the derivative of a product of two functions. This rule can also be used to calculate the ln 8x derivative. The product rule for two functions formula f(x) and g(x) is written as;

d/dx{f(x)g(x)}=f(x)g(x)+g(x)f(x)

The formula of product rule is a fundamental formula in calculus and is used to find the derivatives of many functions, including polynomial functions, trigonometric functions, and exponential functions.

Proof of derivative of ln8x by product rule 

The function ln x can be written as;

f(x)= 1. ln(8x)

Applying derivative with respect to x,

f’(x)=(1. ln(8x))’

Applying product rule, 

f’(x)=1.(ln(8x))’+ln(8x) (0)

f’(x)=1.(1/x)+0

Therefore,

f’(x)=1/x

Hence the derivative of ln(8x) is always equal to the reciprocal of x. Use our product rule calculator to solve derivative of ln8x online.

How to find the derivative of ln8 with a calculator?

The easiest way to calculate the ln8x derivative 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 tool.

1. 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 ln(8x) derivative.
2. Now, select the variable by which you want to differentiate ln(8x). Here you have to choose ‘x’.
3. Select how many times you want to differentiate ln(8x). In this step, you can choose 2 for second, 3 for third derivative and so on.
4. Click on the calculate button. After this step, you will get the differentiation of ln(8x) within a few seconds.

FAQ’s

What is the relation between ln x and log x?

Most of the equations used in mathematics are derived by using calculus. It involves natural logarithms. The relation between ln x and log x is;

ln x = 2.303 log x 

What is the derivative of ln(8x)?

The derivative of ln(8x) can be calculated as;

d/dx (ln(8x)) = 1/x