Calculus, all content (2017 edition)
Course: Calculus, all content (2017 edition) > Unit 2Lesson 28: Logarithmic functions differentiation
- Derivatives of sin(x), cos(x), tan(x), eˣ & ln(x)
- Derivative of logₐx (for any positive base a≠1)
- Worked example: Derivative of log₄(x²+x) using the chain rule
- Differentiate logarithmic functions
- Differentiating logarithmic functions using log properties
- Derivative of logarithm for any base (old)
- Differentiating logarithmic functions review
Derivative of logarithm for any base (old)
An older video where Sal finds the derivative of log_b(x) (for any base b) using the derivative of ln(x) and the chain rule. Created by Sal Khan.
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- I dont understand one thing...if (1/ln b) is a constant, shouldnt its derivative be 0 like all the other constants?(22 votes)
- The derivative of all constants IS ZERO. However, if we directly use that property then, the derivative of the entire expression will turn out to be zero, which would be wrong.
Instead, we use another derivative property i.e.,
d/dx [A*f(x)] = A*d/dx [f(x)] , where A = constant
Now, you might be thinking that if we use the aforementioned property, then when or why do we need this following property-
d/dx[A] = 0
We use this property to find the derivative of a polynomial.
d/dx[x^2 + 2] = 2x + 0 = 2x.
Hope you got it.(40 votes)
- Can anyone tell me why and how is log base b of x = ln x/ln b?
Thanks for your help!(3 votes)
- Please read my comment regarding the question financeBoy, I am assuming you 're having the same problem as him. And his question is top rated so I saw his question first.(1 vote)
- Why is the derivative of lnX = 1/x again?(6 votes)
- Why 1/ln b is a constant?(3 votes)
- 1/ln(b) is a constant because when x changes, the value of that expression stays the same. Its value stays constant.(11 votes)
- Why can you remove a constant from the derivative?(3 votes)
- Because the constant is just a multiplier. It makes no difference when you choose to do the multiplication. Perhaps it would be easier to see with an easier derivative:
Let us find the first derivative of 9x² with respect to x.
We can do this straight up as d/dx (9x²) = (2)(9x) = 18x
Or, we can pull out the constant 9 as in:
d/dx (9x²) = 9 · d/dx (x²) = 9 · (2x) = 18x(8 votes)
- What is the difference between writing "logx" and "lux" ?(2 votes)
- log(x) uses the base 10 while ln(x) called the natural logarithm uses e as the base. The formulas in calculus use the natural logarithm only so the base has to be converted by using the formula Sal mentions at around0:50
Logarithms tell how many time a number(base) has to be raised to for getting another number. Like 10^2 = 100 so log(100) = 2
When log is mentioned without a base it is always 10. The natural logarithm(ln) uses another base called e or Euler's number and is approximately 2.71(8 votes)
- What is the differentiation of log constant with base x with respect to x?(2 votes)
- We have y=log(basex)(c) where c is a constant.
First, we are going to make x be put to both sides.
next, log both sides. yln(x)=ln(c)
divide by ln(x) y=ln(c)/ln(x)
now, take the derivative of both sides (You need the chain rule for this part which you might not know yet. You can always watch a video on it.). dy/dx=ln(c)/(x*ln(x)^2)
so that's what it is, d/dx( log(basex)(c)) = ln(c)/(x*ln(x)^2)
I hope this helped.(3 votes)
- From0:40onwards, I notice that whenever Sal has to find a derivative of a log with an arbitrary base, he always chooses to represent the log (via change of base formula) as a base-e log and not a base-10 log. Is that how you should always interpret a log with an arbitrary base when finding its derivative?(2 votes)
- It is not about interpreting (what ever you meant by that). The only reason, we transform to base-e is simply that we know its derivative. There is (as far as I know) no other way to do it.(2 votes)
- Why EXACTLY can we take 1/lnb out of the derivative? inb4 go see some proof that is much farther down the line of lecture videos. you should probably prove things before you start using them, or people will get lost.(2 votes)
- Yes, any constant can be factored out of a derivative. There is no actual need to do that, but sometimes it is more convenient to do so.(2 votes)
- what if there is a constant in front of the log(2 votes)
We already know that the derivative with respect to x of the natural log of x is equal to 1/x. But what about the derivative, not of the natural log of x, but some logarithm with a different base? So maybe you could write log base b of x where b is an arbitrary base. How do we evaluate this right over here? And the trick is to write this using the change of base formula. So we could write it in terms of logarithms. We know that log-- I'm just going to restate the change of base formula. And I'm going to change from log base b to log base e, which is essentially the natural log. So the change of base formula, we prove it elsewhere on the site. Feel free to search for it on the Khan Academy. The change of base formula tells us that log base b of x is equal to the natural log, if we want to go to log base e. The natural log of x over the natural-- actually let me write it as an explicit logarithm so it makes it clear what I'm doing. Log base e of x over log base e of b, which is the exact same thing as the natural log of x over the natural log of b. So all we have to do is rewrite this thing. This is equal to the derivative with respect to x of the natural log of x over the natural log of b. Or we could even write it as 1 over the natural log of b times the natural log of x. And now this becomes pretty straightforward. Because what we have right here, 1 over the natural log of b, this is just a constant that's multiplying the natural log of x. So we could take it out of the derivative. So this is the same thing as 1 over the natural log of b times the derivative with respect to x of the natural log of x. And we know what to do with this. This thing right over here is just going to be equal to 1/x. So we end up with 1 over the natural log of b times 1/x. So we end up with 1 over the natural log of b times 1/x, or 1 over the natural log of b, which is just a number times x. So if someone asks you what is the derivative with respect to x of log base 5 of x, well, now you know. It's going to be 1 over the natural log of 5 times x, just like that.