Class 12 math (India)
- Derivative of logₐx (for any positive base a≠1)
- Logarithmic functions differentiation intro
- Worked example: Derivative of log₄(x²+x) using the chain rule
- Differentiate logarithmic functions
- Differentiating logarithmic functions using log properties
- Differentiating logarithmic functions review
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:
is a specific case of the general form where . Since we obtain the same result.
You can actually use the derivative of
(along with the constant multiple rule) to obtain the general derivative of .
Want to learn more about differentiating logarithmic functions? Check out this video.
Practice set 1: argument is
Practice set 2: argument is a polynomial
Want to join the conversation?
- I can follow these equation but i can't follow x^x any advice? Using logarithmic differentiation.(0 votes)
- From my understanding, you'd like help with how to differentiate x^x. This is how you do it:
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.(17 votes)
- 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?(2 votes)
- 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.(8 votes)
- 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? 😉.
- Logarithmic differentiation should work here. Can you provide us with your steps so we can perhaps find an error?(2 votes)
- What do you do if the "x" is not simply x, but is raised to a power or if the equation is log base 4 of x-2?(1 vote)
- 1/x(ln(a)) or I could do natural log of y equals (the power- assuming that the power is a variable) times the natural log of x. If the power is a number I would multiply it by the coefficient of x and subtract 1 from the exponent.(0 votes)
- when the differentiation of logarithm is applied in real life?(0 votes)
- 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!(2 votes)
- my mr of calculus said for as that K.A. is not good for us and I tell him that K.A. is better than you .... Am I correct?(0 votes)