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.

## AP®︎/College Calculus AB

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

Lesson 1: Defining average and instantaneous rates of change at a point

# Derivative as a concept

Learn about derivatives as the instantaneous rate of change and the slope of the tangent line. This video introduces key concepts, including the difference between average and instantaneous rates of change, and how derivatives are central to differential calculus. Master various notations used to represent derivatives, such as Leibniz's, Lagrange's, and Newton's notations.

## Want to join the conversation?

• why slope of a line is not x/y?
• There are two natural reasons as to why slope is Δ𝑦/Δ𝑥 instead of the reciprocal.
First, in everyday language, we say that something is steep if it has a large slope such that a small change horizontally corresponds to a drastic (large) change vertically. A mountain is steeper (has a greater slope) if for every 1 meter you go forward your altitude increases by 10 meters than if for every 1 meter you go forward your altitude increases by 2 meters. This everyday definition gives us Δ𝑦/Δ𝑥 for slope.

Also, in terms of a linear equation, 𝑥 is viewed as an independent variable, that is, the variable we have control over. We can set 𝑥 to anything we want. However, 𝑦 is the dependent variable. We have no control over its value – it is completely determined by 𝑥. So it is natural that we would want to gauge how much change in the dependent variable is caused by a unit change in the independent variable because we have control over the independent variable whereas the dependent variable is determined by the independent variable, not by us directly. This notion again gives us Δ𝑦/Δ𝑥 as the slope.
• how could a point have a rate of change ? i mean it is a point a coordinate the change happen when we move from coordinate to other right ?
• Hi Khalid,
In this case we are referring to instantaneous rate of change at the instant we 'get' to that point... the best way to visualize a rate of change at a point is to draw in a tangent line to the curve at that point... the slope of that line is your rate of change of the function at that point.

:)
• what is the difference between `delta` x and dx?
• Δx describes discrete change; i.e., you can say Δx = 1 or 0.1, and is probably used more in algebra.
dx represents an infinitesimal change, i.e., it doesn't have a value like dx = 0.0000001, but is simply infinitesimal (not a very rigorous explanation, I know). It's the calculus counterpart to Δx; because it's infinitesimal, a series of dx's put together can describe continuous change (as with derivatives and integrals).
• How can a single point on a plot dictate a slope of a tangential line? There could be multiple combinations of y-intercepts and slopes with a single point. Can someone explain this to me more?
• For any given point on a curve, there is only one line you can draw that will be tangent to that curve. As you go through and watch more videos, you'll find out how to take the derivative of an equation. When you plug x into that derivative equation, the result you'll get for y (or f(x)) will be your tangent line slope. Hope this helps!
• I understand the concept explained in this video. A question arise now. Consider a graph between distance (in y-axis) and time (in x-axis). Now, if we take a derivative, what we do is that the change in the x value (dx) when dt is realy close to zero (infinitely small). Usually, dx/dt is known as the velocity. Thats Okay. But, how the unit is m/s (meter-per-second) even though we use infinitesimally small time?
• This is informal but let's say the distance changes twice as fast as the time. The ratio is always 2:1, no matter how big or small. The small time is also cancelled out by the small distance.
• Just out of curiosity, what happens if you take the derivative of a function's derivative? Is there a use for that?
• Yes, that's called the second derivative. In fact, it's very useful for finding things like concavity (which way the graph curves). It'll be just a bit farther down the course.
• Consider the graph of y=|x|.
What would the derivative be at x=0? I'm wondering this because it intuitively feels like there should be infinite possible tangent lines to that point. Can a point have more than one derivative?
• Nice question!
You are right that in a sense, this derivative is ambiguous. The derivative of |x| at x=0 does not exist because, in a sense, the graph of y=|x| has a sharp corner at x=0.
More precisely, the limit definition of this derivative is

lim h-->0 of (|0+h|-|0|)/h = lim h-->0 of |h|/h.

Since lim h-->0^+ of |h|/h = lim h-->0^+ of h/h = 1, but
lim h-->0^- of |h|/h = lim h-->0^- of -h/h = -1, we see that
lim h-->0 of |h|/h does not exist.

So this derivative does not exist! Note that this example shows that it's possible for a function to be continuous at a point without being differentiable there.
• so just checking my understanding, in this video, the slope of a secant line is used to calculate the average rate of change and the slope of a tangent line is used to calculate the derivative?

And any relation between the secant with the curve here and the secant we learned in trigonometry?
• Yep! You can also say that the slope of the tangent is used to get the instantaneous rate of change at a point. The "at a point" is important as the secant gives the rate of change "between two points"

There actually is! Check this (https://en.wikipedia.org/wiki/File:Unitcircledefs.svg) and you'll see why the line is called the secant line. You can also see why the tangent line is called so.