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### Course: AP®︎/College Calculus AB>Unit 2

Lesson 8: Derivatives of cos(x), sin(x), 𝑒ˣ, and ln(x)

# 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!
• Hear me out, what if x < 0
(1 vote)
• since the function isnt defined at x < 0 then it isnt differentiable or the derivative doesnt exist at x < 0 (there are no derivatives/tangent lines for x values that dont exist)
so basically the derivative of a function has the same domain as the function itself. Therefore the derivative of the function f(x)= ln(x), which is defined only of x > 0, is also defined only for x > 0 (f'(x) = 1/x where x > 0).
i hope this makes sense
• 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))?