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## AP®︎ Calculus AB (2017 edition)

### Course: AP®︎ Calculus AB (2017 edition)>Unit 2

Lesson 1: The derivative: An intuitive introduction

# Derivative as a concept

AP.CALC:
CHA‑2 (EU)
,
CHA‑2.A (LO)
,
CHA‑2.A.1 (EK)
,
CHA‑2.B (LO)
,
CHA‑2.B.1 (EK)
Introduction to the idea of a derivative as instantaneous rate of change or the slope of the tangent line.

## 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!
• 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.
• 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.
• When he says the change in x getting ever closer to 0, is he referring to the fact that as we approach our desired point, there is less and less change in the x value?
• Essentially, yes. The less change in the x-value, the more accurate the slope is at the desired point.
• Is the secant he is referring to the one defined as a line that intersects 2 points on a curve?
• Yes, a secant line is a line that intersects a curve at a minimum of two distinct points.
• Hey everyone...so I have a question that may be more 'semantics'. But when I am getting more into physics I see a lot of talk about "Deriving an equation"...or something like "Derive the kinematic equations" etc... Is this the same meaning as literally taking the derivative of the equation? Or does "deriving" in this sense mean something else?
• They are two different meanings.

Deriving an equation in physics means to find where an equation comes from. It is somewhat like writing a mathematical proof (though not as rigorous).

In calculus, "deriving," or taking the derivative, means to find the "slope" of a given function. I put slope in quotes because it usually to the slope of a line. Derivatives, on the other hand, are a measure of the rate of change, but they apply to almost any function. Think of them as an extension of the concept of slope.