Proof of Derivative of Natural log x
There are two methods to find the derivative of Natural log x:
- Using First Principle of Differentiation
- Using Implicit Differentiation
Derivative of Root x by First principle of differentiation
First principle of differentiation states that derivative of a function f(x) is defined as,
f'(x) = limh→0 [f (x + h) – f(x)] / [(x + h) – x]
or
f'(x) = limh→0 [f (x + h) – f(x)]/ h
Putting f(x) = log x in the above equation, we get,
f'(x) = limh→0 [log (x + h) – log(x)]/ h
Using the property of logarithmic functions, i.e. log a – log b = log(a/b), we get,
⇒ f'(x) = limh→0 [log (1 + h/x)]/ h
Multiplying 1/x with numerator and denominator, we get,
⇒ f'(x) = limh→0 (1/x) * [log (1 + h/x)]/(h/x)
Using the standard result of limits, we have,
⇒ limh→0 [log (1 + h/x)]/(h/x) = 1
Thus,
⇒ f'(x) = limh→0 (1/x) *(1)
⇒ f'(x) = 1/x
Hence, we have derived the derivative of natural log x by using first principle of differentiation.
Derivative of Natural log x by using Implicit Differentiation
Implicit differentiation is a process of differentiation in which a function y = f(x) is expressed as x = f(y), where f(y) is such a function whose derivative is a standard result or is easier to calculate. Let us take a look on how it can be used to find derivative of natural log x.
Let, y = log x
⇒ ey = x
Differentiating on both sides we get,
⇒ eydy = dx (∵ Derivative of ey = ey )
⇒ dy/dx = 1/ey
⇒ dy/dx = 1/x
Thus, we have derived formula for derivative of natural log x using implicit differentiation.
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Derivative of ln x (Natural Log)
The derivative of ln x is 1/x. We can also say that the derivative of natural log x is 1/x. The derivative of any function gives the change in the functional value with respect to change in the input variable. Natural log x is an abbreviation for the logarithmic function with the base as Euler’s Number, i.e. e.
In this article, we will discuss the derivative of natural log x, various methods to derive it including the first principal method and implicit differentiation, some solved examples, and practice problems.