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# The derivative of x² at x=3 using the formal definition

AP.CALC:
CHA‑2 (EU)
,
CHA‑2.B (LO)
,
CHA‑2.B.2 (EK)
,
CHA‑2.B.3 (EK)
,
CHA‑2.B.4 (EK)
Sal finds the limit expression for the derivative of f(x)=x² at the point x=3 and evaluates it. Created by Sal Khan.

## Want to join the conversation?

• I don't understand why a derivative is written as F'(x). Why is that the notation we use? Thanks for any help! •   Simple notation:
1. Lagrange introduced the prime notation f'(x). We use it because is one of the most common modern notations and is most useful when we wish to talk about the derivative as being a function itself.
2. Newton introduced the dot notation ẏ, used in physics to denote time derivatives.
3. Leibniz introduced the Leibniz's notation dy/dx, useful for partial differentiation, and
4. Euler introduced the Euler's notation which uses a differential operator Df(x) useful for solving linear differential equations.
• Aren't we looking for the limit as x approaches 3 ??? • What is the use of a derivative? Any real-life applications? • The most obvious application quoted is usually for Speed, Acceleration and Distance. They are all derivatives/integrals of each other. E.g. Acceleration is the derivative of Speed.

However there are any number of possibilities since it can be applied to pretty much anything where an analysis of a rate of change is helpful. Actually calculating a best fit function in the first place is an expertise in itself though.
• At , Sal simplifies (6(delta)x + (delta)x^2)/(delta)x to 6+(delta)x. However, later he substitutes (delta)x for 0 to get a slope of 6. If he'd put that in the original expression it would have been division by 0. Why does this work? Aren't you not allowed to divide by 0? • so a derivative is a function that shows the slope of the original function (curve) ? • I keep seeing the term "limit of difference quotient" in my worksheets. Is this just another term for what this video uses or is it an entirely separate concept? • Indeed it is the same thing. The difference quotient (in calculus) is used to refer to the slope between two points, the average rate of change between two points, rise over run, ∆y/∆x, or whatever other name you might have for it. The limit of the difference quotient then becomes the instantaneous rate of change aka the derivative.
• Hey, we can solve a curve using a tangent line. Why use a secant line then? • So we can write f'(x) as dy/dx? • yep another way of writing it is d/dx y or the derivative of y with respect to x.

I say that because when you take the derivative of the derivative it can be written as f''(x) or (d^2)y/(dx)^2 because basically you are doing d/dx d/dx y. It might look like things are being multiplied and you'll see that kind of notation used a lot in calculus.

Hopefully that last part made sense, but if not, to make sure I answer your question, Yes those are two ways of writing the derivative.
• Is there also a tangent line on a graph which is periodical? Like cos(x) or sin(x) for example? • Absolutely. For any f(x), determining f'(x) gives the slope of the tangent line at that point. So there are actually an infinite number of tangent lines on a sinusoidal function. The tangent line at f(0) on a graph of f(x)=cos x would have a slope of zero, and would just be the line y=1. That same line would also be the tangent at f(2pi). At f(pi), the slope is also zero, but at a minima. So there the tangent line is y=(-1). In between the minima and maxima (high and low points where the slope=0) there are an infinite number of tangent lines as the slope continuously changes. Graph a sin wave and move a ruler along it as a tangent line, or take two straightedges and "sandwich" the wave from above and below. Those are both tangent lines. It may help you see it. I know there are many programs that illustrate the concept but, for me anyway, sometimes holding a physical representation helps. `( 6∆x + (∆x)^2 ) / ∆x = 6∆x/∆x + (∆x)^2/∆x`
`6 + ∆x`