**Introduction** to the derivative of ln7x

Derivatives have a wide range of applications in almost every field of engineering and science. The derivative of ln(7x) can be calculated by following the rules of differentiation. Or, we can directly find the ln7x derivative by applying the first principle of differentiation. In this article, you will learn what the derivative of ln 7x is and how to calculate the ln(7x) derivative by using different approaches.

**What is the derivative of ln(7x)?**

The derivative of ln7x with respect to x is equal to 1/x. This can be expressed as d/dx ln(7x). It represents the rate of change of the natural logarithmic function ln(7x).

Mathematically, the function ln(7x) is written as:

$\ln 7x=\log_e 7x$

The expression loge(7x) represents the logarithm of 7x with base e. This is a useful way to express the natural logarithm of 7x, because it helps us to easily calculate its derivative and understand its relationship with other mathematical functions.

**Differentiation of ln(7x) formula**

The differentiation of ln(7x) is equal to 1/x. It can be calculated the same as the derivative of ln(x). Applying this formula to ln(7x), you can simplify it to:

$\frac{d}{dx}(\ln 7x) =\frac{1}{x}$

This means that the derivative of ln 7x is equal to the reciprocal of x. It means that the rate of change of the ln(7x) 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 ln(7x) derivative?**

There are various derivative rules ways to derive derivatives of ln(7x). Therefore, we can prove the derivative of ln(7x) by using;

- First Principle
- Implicit Differentiation
- Product Rule

**Derivative of ln 7x by first principle**

The first principle of differentiation tells us that to find the derivative of ln(7x), 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(7x), 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 of ln 7x by first principle**

To differentiate ln7x by using first principle, we start by replace f(x) by ln 7x. Also, replace f(x) by ln(5x) to find the ln(5x) derivative by using this method.

f’(x)=lim{ln7(x+h)-ln(7x)/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)^1/t

Hence by limit formula, we know that,

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

Therefore, the derivative of ln7x is;

f’(x)=1/x

**Derivative of ln(7x) 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(7x).

**Proof of derivative ln(7x) by implicit differentiation **

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

y=ln(7x)

Converting in exponential form,

ey = 7x

Applying derivative on both sides,

d/dx(ey)=d/dx(7x)

ey.dy/dx = 7

Now,

dy/dx=7/ey

Since x=ey

Therefore,

dy/dx=1/x

To find the differentiate ln(7x) online, use our implicit differentiation calculator online.

**Derivative of ln7x 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 7x derivative. The product rule for two functions calculator 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 ln(7x) by product rule **

The function ln 7x can be written as;

f(x)= 1. ln(7x)

Applying derivative with respect to x,

f’(x)=(1. ln(7x))’

Applying product rule,

f’(x)=1.(ln(7x))’+ln(7x) (0)

f’(x)=1.(1/x)+0

Therefore,

f’(x)=1/x

Hence the derivative of ln7x is always equal to the reciprocal of x. Also find how to evaluate derivative of ln(e^x) by using product rule.

**How to find the differentiation of ln(7x) with a calculator?**

The easiest way to calculate the ln(7x) 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.

- 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 ln 7x.
- Now, select the variable by which you want to differentiate ln7x. Here you have to choose ‘x’.
- Select how many times you want to differentiate ln(7x). In this step, you can choose 2 for second, 3 for third derivative and so on.
- Click on the calculate button. After this step, you will get the derivative of ln(7x) within a few seconds.

**FAQ’s**

**What is the derivative of e****2****?**

The function e2 is an exponential function. But it does not include any variable therefore, it is a constant function. Since the derivative of a constant is zero. Therefore the derivative of e2 is zero.

**What is the derivative of ln(7x)?**

The derivative of ln(7x) can be calculated as;

d/dx (ln(7x)) = 1/x