I have a natural logarithm with e^x/1+e^x. I separated it with the log rules but then I'm stuck. Any advice?

i think you are asking about finding d/dx( ln( e^x / 1 + e^x) ). so im solving for that and here it is: we can write ==> ln(e^x / 1+e^x) as ln(e^x) - ln(1+e^x) so now when we differentiate we can differentiate them independently. so d/dx( ln( e^x / 1 + e^x) ) = d/dx( ln(e^x) ) - d/dx( ln(1+e^x) ) = ( (1/e^x) *e^x ) -( ( 1/(1+e^x) ) * e^x ) let me know if we have any confusion.

I can follow these equation but i can't follow x^x any advice? Using logarithmic differentiation.

From my understanding, you'd like help with how to differentiate x^x. This is how you do it: y=x^x Take the logs of both sides: ln(y) = ln(x^x) Rule of logarithms says you can move a power to multiply the log: ln(y) = xln(x) Now, differentiate using implicit differentiation for ln(y) and product rule for xln(x): 1/y dy/dx = 1*ln(x) + x(1/x) 1/y dy/dx = ln(x) + 1 Move the y to the other side: dy/dx = y (ln(x) + 1) But you already know what y is... it is x^x, your original function. So sub in: dy/dx = x^x(ln(x) + 1) And you're done.

Are there “rules” for when you can(not) use logarithmic differentiation (including implicit)? I ask because of the following KA problem: “Find dy/dx for x=√(xy+1)” For that problem I attempted to immediately use logarithmic differentiation, e.g. ln(x)=ln(√(XY+1)). However having now worked on it a good deal I have come to understand that logarithmic differentiation generates an incorrect result. Why doesn’t logarithmic differentiation work in this case? (I speculate that perhaps it is because there is a single term that has more than one variable – e.g. XY messes it up – but that is just a guess). Note that the following answer is not sufficient: “You shouldn’t use logarithmic differentiation on that problem.” E.g. I (now) understand it won’t work - I want to know WHY it doesn’t work - what is the rule I should use so that I don't try to do that again in the future? 😉. Thanks, kevin

Logarithmic differentiation should work here. Can you provide us with your steps so we can perhaps find an error?

when the differentiation of logarithm is applied in real life?

There are almost "never-ending applications".....I will give you application which is a compelling and spectacular one (unfortunately in negative terms)....Measuring the size of earthquakes requires the knowledge of logarithmic functions!

Main content

Class 12 math (India)

Course: Class 12 math (India) > Unit 5

Lesson 15: Logarithmic functions differentiation

Differentiating logarithmic functions review

Google Classroom

Review your logarithmic function differentiation skills and use them to solve problems.

How do I differentiate logarithmic functions?

First, you should know the derivatives for the basic logarithmic functions:

\frac{d}{d x} \ln (x) = \frac{1}{x}

\frac{d}{d x} \log_{b} (x) = \frac{1}{\ln (b) \cdot x}

Notice that

\ln (x) = \log_{e} (x)

is a specific case of the general form

\log_{b} (x)

where

b = e

. Since

\ln (e) = 1

we obtain the same result.

You can actually use the derivative of

\ln (x)

(along with the constant multiple rule) to obtain the general derivative of

\log_{b} (x)

Want to learn more about differentiating logarithmic functions? Check out this video.

Practice set 1: argument is $x$ ‍

Problem 1.1

h (x) = 7 \ln (x)

h^{'} (x) = ?

Want to try more problems like this? Check out this exercise.

Practice set 2: argument is a polynomial

Problem 2.1

g (x) = \ln (2 x^{3} + 1)

g^{'} (x) = ?