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## AP®︎ Calculus AB (2017 edition)

### Course: AP®︎ Calculus AB (2017 edition)>Unit 4

Lesson 7: Derivatives of logarithmic functions

# Derivative of ln(x)

The derivative of ln(x) is 1/x. We show why it is so in a different video, but you can get some intuition here.

## Want to join the conversation?

• But ln(x) is a logarithmic function defined only for x-values greater than zero, while 1/x is a rational function defined for all non-zero x's. So would it be more accurate to say: the derivative of ln(x) is 1/x such that x is greater than zero?
• Really good thinking here, but since the domain is already limited with ln(x) when we start, we don't need to carry that over, since we already know x can't be 0 or less. if this were the other way around , where we started with a larger domain we would have to do something to the domain of the derivative.

Spoiler alert, this happens when you get to integrals.
• what happens when you have an equation like 3ln(x) would it be 3/x ?
• That's correct, because we leave scaling constants alone when taking derivatives.
• Can we say that if x--> inf
the slope is 0 and vice- versa. This might seem foolish but Im just curious
• What is the derivative of 2x?
• The derivative of a function is its slope.
y=2x is a line of slope 2.
So the derivative of 2x is 2.
• Hi, what happens when we have to do ln 4x - is it 1/4x?
(1 vote)
• No, actually. To find the derivative of ln(4x), you have to use the chain rule.

ln(4x) = 1/(4x) * 4 = 1/x

Hope this helps!
• I got a question asking "d/dx ln(2-e^x)". I thought the answer was "1/(2-e^x)" but the answer was
"(1/(2-e^x))(-e^x)". Can someone explain?
(1 vote)
• "Slope of a tangent line" and the derivative are the same thing, right?

It's like if you take a chunk from the original function's line at a certain point, same sized bit on either side of the line and "straighten it out", you get the tangent line and the derivative of that point, an m value. Right?
(1 vote)
• Be careful with your wording. However, I think you got right idea
(1 vote)
• What is the derivative of ln(f(x))?
• This is an example of a composite function. A composite function like g(f(x)).
The differentiation of composite functions is done using the chain rule. This will be covered in the next modules but for now the differentiation of
d/dx(ln(f(x))) = 1/f(x)*f'(x)