Derivative of ln x (Natural Log)
Last Updated : 21 Aug, 2024
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 the 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.
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What is Derivative of Natural log x?
Derivative of Natural log x is 1/x. It implies that change in the value of log x with respect to change in the input variable, i.e. x is 1/x. Also, it defines the slope of the tangent to the curve represented by y = log x, at any point x = x1. The formula for derivative of natural log x is written as follows:.
Derivative of Natural log x Formula
d/dx [log x] = 1/x
or
(log x)' = 1/x
The derivation for this formula using the first principle of differentiation and implicit differentiation is discussed as follows.
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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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Examples on Derivative of Natural log x
Example 1: Find the derivative of the function represented as f(x) = log(x2+4x+5).
Solution:
We have, f(x) = log(x2+4x+5)
By applying chain rule, we get,
⇒ f'(x) = [1/(x2+4x+5)] × d/dx(x2+4x+5)
⇒ f'(x) = [1/(x2+4x+5)] × (2x+4)
⇒ f'(x) = (2x+4)/(x2+4x+5)
Example 2: Find the derivative of the function given by f(x) = 2√(log x).
Solution:
We know that (√x)' = 1/2√x and (log x)' = 1/x
For f(x) = 2√ (log x), by applying chain rule, we get,
⇒ f'(x) = 2×[1/2√ (log x)] × d/dx (log x)
⇒ f'(x) = [1/√ (log x)] × (1/x)
⇒ f'(x) = 1/x√ (log x)
Example 3: If a curve is represented as y = log √x, derive an expression for dy/dx.
Solution:
We know that dy/dx is simply the derivative of the function represented by y = f(x).
Therefore, by chain rule,
For y = log √x
⇒ dy/dx = (1/√x) × (1/2√x)
⇒ dy/dx = 1/2x
Thus, for y = log √x, we get dy/dx = 1/2x.
Example 4: Find an expression for slope of the tangent to the curve represented by y = (log x)/x.
Solution:
We know that slope of tangent to the curve is given by the derivative of the function represented as y = f(x). Thus, we need to calculate dy / dx for y = (log x)/x.
By applying quotient rule, we get,
⇒ dy / dx = [x × d/dx (log x) - (log x) × d/dx(x)]/x2
⇒ dy/dx = (1 - log x) / x2
Thus, slope to the tangent of the curve represented by y = (log x)/x is given as (1 - log x)/x2.
Example 5: If f(x) = sin (log x), determine the expression for f'(x).
Solution:
We know that (sinx)' = cosx and (log x)' = 1/x,
Thus, for f(x) = sin (log x), applying chain rule, we get,
⇒ f'(x) = cos (log x) × (log x)'
⇒ f'(x) = cos (log x)/x
Hence, for f(x) = sin (log x), we have, f'(x) = cos (log x)/x
Practice Questions on Derivative of Natural log x
Q1. If y = x/log x, then find the value of dy/dx.
Q2. If y = log x/sinx, find the value of dy/dx.
Q3. Find the derivative of the function f(x) = log(x2+3x+4).
Q4. Find the derivative of the function f(x) = √(log x).
Q5. Find the value of f'(x), if f(x) = x log x.
Q6. Find the derivative of g(x)=ln(cos(x)).
Q7. Differentiate h(x)=ln(tan(x)).
Q8. Calculate the derivative of y(x)=ln(x2+1).
Q9. Find the derivative of f(x)=ln(sin(x)+cos(x)).
Q10. Differentiate g(x)=ln(2x+3).
Summary
The derivative of the natural logarithm function ln(x)with respect to xxx is given by \frac{1}{x} . This result holds for all x>0. In other words, if you have a function f(x)= ln(x), then its derivative f′(x)\frac{d}{dx}[\ln(x)] = \frac{1}{x}. This derivative tells us how the rate of change of ln(x) varies with x, and it is a fundamental result in calculus used in various applications, including solving differential equations and optimizing functions.
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