If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Secant line with arbitrary difference

Sal finds the slope of the secant line on the graph of ln(x) between the points (2,ln2) and (2+h,ln(2+h)).

## Want to join the conversation?

• Clarification of terms requested. At , Sal refers to slope of the secant line as delta y / delta x (rise / run). The line is highlighted in blue. Is not rise over run the same as opp / adj and isn't this the tangent, not the secant? And the blue highlighted line, isn't that the hypotenuse, not the secant. I thought the secant is the inverse of the cosine, or hyp / adj. Thanks for the answers.
• What is arbitrary difference?
• "Arbitrary" means "chosen in no particular way". So if we have an arbitrary number, x, between 0 and 1, that means that x can be any number between 0 and 1.

Arbitrariness is important in math, especially with proofs, because if we prove that (for instance) some property holds with an arbitrary distance, then we know that the property holds for any distance at all.
• Dears, I am confused a little. Are these two "guys" the same:
- "secant line"
and
- "slope of secant line"
or not?
• The secant line is a geometric construction. It's your normal concept of a line, drawn between two points.

The slope of the secant line is a real number. It describes how tilted the secant line is.
• What does arbitrary difference or arbitrary point mean? It is in the title but not in video.
• Arbitrary in this case is used to mean "as close as you want it to be," which is extremely important when we're discussing tangents to the curve and a derivative of the function at a point.
• Doesn't "h" have to be specified to be a number greater than -2?
• Strictly speaking I would say you are correct – in the context of derivatives h is usually thought of as being "arbitrarily" small (i.e. close to zero), which is why Sal probably didn't think (or bother) to mention that ...
• what is the difference between ln(x) and log(x)?
(1 vote)
• Couldn't you further simplify to ln(h) / h? from ln(2) + ln(h) - ln(2) / h
(1 vote)
• a normal formula could solve all these kinds of question