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

Discover how to find the derivative of x² at x=3 using the formal definition of a derivative. Learn to calculate the slope of the tangent line at a specific point on the curve y=x² by applying the limit as the change in x approaches zero. This method helps determine the instantaneous rate of change for the function. Created by Sal Khan.

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• 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 ???
• No, because we want to find the derivative, or the limit as Δx approaches zero.
• 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?
• That's not exactly what he does. He makes dx very very very small, but not zero. Close to zero, but not zero.

6+0.0000000000000000000000001 is pretty close to 6, and the smaller you make dx the less significant it gets. It will not become exactly zero, but it will become so small that we can ignore it.
• so a derivative is a function that shows the slope of the original function (curve) ?
• Exactly!
If f'(x) is the derivative of the original function f(x).
Then, as you said, f'(x) would be the slope of the function f(x) at the point x.
For example:
f'(5) = 2 would mean that the slope of f(x) at the point x=5 is 2.
• 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?
• I think it's because we get the idea of a tangent line from a secant line which intersect with the curve.
• 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.