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SAT Derivative of Log x Definition, Formula, Proof with Solved Examples

Last Updated on Mar 03, 2025
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What is Derivative of Log x?

Derivative of is . Here “” is the derivative of “”. “” is called the natural logarithm or it is a logarithm with base ‘’, i.e. .

In general, a logarithm has the form . That is, we call the base of the logarithm. Also, represents the number we raise to in order to get . Now, notice that doesn’t have a base shown. When this is the case, the implied base is . Therefore, . So the derivative of is .

The derivative of is denoted by or . Thus, we have

  • or = .
  • or = .

Derivative of Log x Proof by First Principle

To Prove: by using first principle.

Proof: Let

By first principle, the derivative of a logarithmic function (which is denoted by is given by the limit,

Since , we have .

Substituting these values in the equation of first principle,

By using the property of logarithms, , we get

Assume , we get

When ,

Then the above limit becomes

By using property of logarithm, , we get

By using a property of exponents, , we get

Here, the variable of the limit is ‘’. So we can write outside of the limit.

Using one of the formulas of limits, .

[’ and ‘’ are interchanged]

[

Therefore, the derivative of is by using first principle.

Derivative of Log x Proof by Implicit Differentiation

To Prove: by using implicit differentiation.

Proof: Let

Taking the derivative on both sides with respect to , we get

 

By using the chain rule, we get

Therefore, the derivative of is by using implicit function.

Derivative of Log x Proof using Derivative of ln x

To Prove: by using the derivative of .

Note that the derivative of is . We can convert ‘’ into ‘’ using change of base rule.

Proof: Let

By using change of base rule, we have

We know that . Thus, we get

Now find the derivative of , then we get

We know that , we get

Therefore, the derivative of is by using derivative of .

Solved Examples of Derivative of Log x

 

Example 1: Find the derivative of .

Solution: Let

Using the derivatives of formula, we have

By using chain rule, we get

Hence, the derivative of is .

Example 2: Find the derivative of .

Solution: Let

Using the derivatives of formula, we have

By quotient rule, we have

Hence, the derivative of is .

We hope that the above article is helpful for your understanding and exam preparations. Stay tuned to the Testbook App for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.

Derivative of Log X FAQs

The derivative of with base is .

The formulas for derivative of log are given as: or = . or = .

The second order derivative of is

The default base of is .

The derivative of with base is . Hence, the derivative of with base is

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