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.

# Formal definition of the derivative as a limit

Discover how to define the derivative of a function at a specific point using the limit of the slope of the secant line. We'll explore the concept of finding the slope as the difference in function values approaches zero, represented by the limit of [f(c)-f(c+h)]/h as h→0. Created by Sal Khan.

## Want to join the conversation?

• I know this concept is important in math, but how is it useful? •   First of all, there are numerous applications.
For example, you could find the rate at which a chemical reaction is happening.
You could maximize the profit made by a company, given functions for demand and price.

Many topics covered in Algebra can become even broader and more specific.
While creating graphs, you can find the maximums and minimums of the function and where the function will increase or decrease and by what rate.

Secondly,
in calculus, the concept of derivatives will be used with the concept of integrals (anti-derivatives).
Integrals also have numerous applications, such as finding the volumes and surface areas of solids.

I cannot cover all of the applications and uses of derivatives in this one answer box, but calculus can be and is applied everywhere you look.
Trust me.
• This almost sounds ridiculous asking this in the calculus playlist, but why is Sal subtracting the point of lesser value from the point of greater value i.e.: [f(x0 +h)-f(x)] why that order? I realize he is applying the slope formula from Algebra, but I've forgotten (if that makes any sense) why we would subtract the points in that order. Well, thanks to anyone who answers this silly and rather minor question XD. •   (x₁ - x₂)/(y₁ - y₂)
= (-1(-x₁ + x₂))/(-1(-y₁ + y₂))
= (x₂ - x₁)/(y₂ - y₁)

So it doesn't matter what order you use :)
• At Sal said "secant line" he also referred to a tangent line, Do these lines have anything to do with the trig functions tangent and secant? •   It would be clearer to say that both of those uses go back to the definition of tangents and secants in circles. That is, a tangent is a line that meets a circle in exactly one point and a secant is a line that intersects a circle in two points, just like it is for an arbitrary curve in calculus.

Here's a quickie program I drew up that illustrates why two of the trig functions got named after the tangent and secant lines of a circle. Hope you enjoy it!

• Why did he use a secant line? When do you use secant versus tangent lines? •  I can answer your second question, secant lines are used when you are given 2 points on a curve and you just find the slope. This slope of those points is average slope of the curve aka mSec. The tangent line is used when you only have one point to get the tangent slope of the curve.
• So, does that mean a curve is made up of tangent lines that are infinitly close to each other? • At he takes the limit as h approaches 0. Why doesn't he take the limit as h approaches x • At , when you take the limit of this function, are you trying to find the slope of the tangent line? Is that why h yields to zero? • Okay I understand the video but I just tried some of the practice problems and I don't understand how I'm supposed to find the derivative of this f(x)=6x4−7x3+7x2+3x−4 using that formula he gave us. • Why to indicate a derivative is used the notation d/dx ?
And can this notation be considered as a ratio of "d" over "dx" , or is just a symbol? • It is a symbol, though there are good reasons that you will learn about later for depicting it is a fraction. But it is not a ratio in the sense you mentioned.

For the moment, though, you should regard d/dx as being a symbol that means "the derivative with respect to x of the following function".
• How would you use this to find the perpendicular tangent line to a function? 