- #1
kittybobo1
- 11
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Homework Statement
I had a quick question concerning some definitions. What is the difference between lim_{x \rightarrow x_0} f'(x) to exist and for f'(x_0) to exist, definition wise?
The derivative of a function is a measure of how that function changes as its input changes. It is the slope of the tangent line to the function at a given point.
The derivative can be calculated using the limit definition: f'(x) = lim(h→0) [(f(x+h) - f(x))/h]. This is also known as the difference quotient.
The derivative of a function at a specific point represents the instantaneous rate of change of that function at that point. This means it tells us how much the function is changing at that exact moment.
Derivatives are used to find maximum and minimum values of functions, which is important in optimization problems. They are also used to model and analyze rates of change in real-world situations, such as in physics and economics.
Yes, derivatives can be applied to any function, including non-linear functions. They can also be used to find the slope of a curve at any point, not just for straight lines.