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AznBoi
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Where are there so many different ways of writing the derivative of the function? Which one is the most commonly used by college professors and that looks the coolest? Probably [tex]\frac{dy}{dx}[/tex] right??
So just using f(x) and x gives you a general equation for the entire graph? That's interesting I've never though about it that way before. Is that the only reason for being able to use it to find the slope at any point on the graph?Gib Z said:Well the limit gives the slop at point (x, f(x) )...and for any point of the corresponding graph, those are the points...you keep it general to all values of x by keeping it as a variable, x.
Gib Z said:Yes. Say we made the limit expression less general. We didn't want the derivative at any point x, but say x=3.
Then the expression becomes
[tex]\lim_{h\to 0} \frac{ f( 3+h) - f(3)}{h}[/tex] which only becomes the derivative at 3.
Just keeping it general allows it to be used for all values of x.
Here's an example: A quadratic ax^2+bx+c is general, which 3x^2-3.34b + pi is a more specific one.
AznBoi said:do you need to use the limit process to find derivatives every time?
Your calculus class teaches you many facts about derivatives. All of those facts are useful.AznBoi said:oh yeah my teacher said that we will be learning shortcuts at the end of the calculus course. Basically you have to do everything the hard way before you can learn the easy way. What if I learn all of the shortcuts first? Would it be bad for me?
Derivative notation is a mathematical notation used to represent the rate of change of a quantity with respect to another quantity. It is commonly used in calculus to calculate the slope of a curve at a specific point.
There are three main varieties of derivative notation: the Newtonian notation, the Leibniz notation, and the Lagrange notation. The Newtonian notation uses a dot above the variable to represent the derivative, while the Leibniz notation uses a prime symbol. The Lagrange notation uses a single quote mark after the variable.
Derivative notation is used in various fields, including physics, economics, and engineering, to model and analyze the behavior of systems that change over time. It is also used in finance to calculate the rate of return on investments.
Derivative notation is a fundamental concept in calculus and is widely used in higher-level mathematics. It is also commonly used in physics and other sciences that involve modeling and analyzing changing systems.
Some common mistakes when using derivative notation include forgetting to use the chain rule, mixing up the independent and dependent variables, and not properly simplifying the derivative expression. It is important to carefully follow the rules of derivative notation and double-check calculations for accuracy.