Punkyc7 said:does it violate the Archimedian property? I'm not sure how to use the definition because how do you expand (x+h)^(1/3)-(x)^1/3. I can't use the binomial theorem to expand it out because that only works for whole numbers.
The derivative of a function is the instantaneous rate of change of that function at a specific point. It represents the slope of the tangent line to the curve at that point.
To find the derivative using the definition, you must first write out the limit expression for the derivative, which is (f(x+h) - f(x)) / h. Then, substitute the given function for f(x) and simplify the expression to find the derivative.
For example, if we have the function f(x) = x^2, the derivative would be (f(x+h) - f(x)) / h = ((x+h)^2 - x^2) / h = (2xh + h^2) / h = 2x + h. As h approaches 0, the derivative becomes 2x, which represents the slope of the tangent line at any point on the curve.
Finding the derivative using the definition is important because it allows us to calculate the slope of a curve at any point, which is useful in many real-world applications such as physics, engineering, and economics. It also helps us understand the behavior of a function and its rate of change.
Yes, there are other methods for finding the derivative, such as using the power rule, product rule, quotient rule, and chain rule. These methods are often quicker and more efficient than using the definition, but they all rely on the same fundamental concept of finding the instantaneous rate of change of a function at a specific point.