Derivatives of natural logarithmic functions

Click For Summary

Discussion Overview

The discussion revolves around the derivatives of natural logarithmic functions, specifically focusing on the differentiation of expressions involving logarithms of rational functions. Participants explore various approaches to deriving these functions and clarify the application of the chain rule and properties of logarithms.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents a derivative of a logarithmic function and questions its correctness.
  • Another participant suggests that the base of the logarithm should be considered, indicating that ln(e) = 1 and discussing the derivative of log(x) in relation to ln(10).
  • A participant proposes a different function and its derivatives, seeking validation of their correctness.
  • Another participant questions the correctness of a derivative expression and asks for clarification on the derivative of ln(12x-5).
  • One participant expresses concern that another is guessing rather than applying systematic methods, suggesting two approaches: using the chain rule or applying properties of logarithms to differentiate.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correctness of the derivative calculations. Multiple competing views and methods for differentiation are presented, and the discussion remains unresolved.

Contextual Notes

Participants express uncertainty regarding the application of the chain rule and the properties of logarithms, with some steps in the differentiation process remaining unclear or contested.

fr33pl4gu3
Messages
82
Reaction score
0
f(x) = ln (12x-5/9x-2)

f'(x) = (4/3) (1/ln10)(9x-2/12x-5)

Is this correct??
 
Physics news on Phys.org
Instead of ln(10) you should put ln(base of the logarithm), in this case; ln(e) = 1. When deriving log(x) (base 10), you can rewrite to ln(x)/ln(10). ln(10) is just a constant, so the derivative of this is 1/ln(10) * 1/x.

Also, I think you didn't do the chain rule quite right.
 
Then, is this
f(x) = ln (12x-5)
g(x) = ln (9x-2)
f'(x) = 12/12x-5
g'(x) = 9/9x-2

Correct??
 
Is this correct:

(12/12(ln9x-2)-5)-(9/9x-2)
 
Last edited:
fr33pl4gu3 said:
(12/12(ln9x-2)-5)-(9/9x-2)
This should not have ln inside. What is d/dx ln(12x-5) ? You've got it from above, just put it into this one.
 
I have the feeling you are guessing more than working systematically. Actually there are two ways to solve this. One is by using the chain rule:
set y = (12x - 5) / (9x - 2). Then the derivative of ln(y) is 1/y . dy/dx.

The other way is to first use ln(a/b) = ln(a) - ln(b) and then use the sum rule to differentiate (you still need the chain rule!).
 

Similar threads

Undergrad Can I Differentiate Both Sides to Solve y in y^2=ln(1-y^2)?
  • · Replies 2 ·
Replies
2
Views
3K
MHB 2.8.218 AP Calculus Exam f'(g(3))=
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Solution to PDA: Solving 12x3 + 6x2 + 9x
  • · Replies 1 ·
Replies
1
Views
2K
Finding where this function is increasing or decreasing
  • · Replies 10 ·
Replies
10
Views
2K
Undergrad Green's function for 2-D Laplacian within square/rectangular boundary
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Finding Max and Min Extremes of a Function with Second Derivatives Equal to Zero
  • · Replies 1 ·
Replies
1
Views
2K
High School How to express ##3^{\sqrt(2)}## in terms of natural logarithms
  • · Replies 10 ·
Replies
10
Views
2K
Solve the given problem giving your answer as a single logarithm
  • · Replies 12 ·
Replies
12
Views
4K
Undergrad Thermodynamics Differentiating logarithms
  • · Replies 11 ·
Replies
11
Views
2K
MHB Inequality solve (x+1)/6<x-(3x-2)/4
  • · Replies 2 ·
Replies
2
Views
2K