Calculus: What is the derivative of ##\phi## at a point?

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Thread starter ElieQuebec10
Start date Oct 12, 2020
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Calculus Derivative Point

In summary, a derivative in calculus is a concept that describes the rate of change of a function at a specific point. It is calculated using the limit definition of the derivative and is closely related to rates of change in real-world scenarios. The notation used for derivatives is dy/dx or f'(x) and they have many applications in fields such as physics, engineering, economics, and statistics for modeling and analyzing real-world situations involving rates of change.

Oct 12, 2020

ElieQuebec10

TL;DR Summary: Can PLEASE someone help me!! I don’t understand it. (I attached a picture)
Thanks!!!!

Last edited by a moderator: Oct 12, 2020

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Oct 12, 2020

fresh_42

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Soit ##\phi: \longrightarrow \mathbb{R}## makes no sense. And as long as ##f## isn't defined, you can only calculate the derivative under the condition that it exists for ##f## and in dependency of ##\nabla f## in which case you have to use the chain rule.

1. What is a derivative in calculus?

A derivative in calculus is a measure of the rate of change of a function with respect to its independent variable. It represents the slope of the tangent line to the function at a specific point.

2. How do you find the derivative of a function?

To find the derivative of a function, you can use the derivative formula, which involves taking the limit of the difference quotient as the change in the independent variable approaches zero. Alternatively, you can use differentiation rules, such as the power rule, product rule, and chain rule, to find the derivative of more complex functions.

3. What does the derivative of a function tell us?

The derivative of a function tells us the rate of change of the function at a specific point. It can also tell us the slope of the tangent line to the function at that point, as well as the direction of the function's change (increasing or decreasing).

4. What is the derivative of ##\phi## at a point?

The derivative of ##\phi## at a point is the instantaneous rate of change of ##\phi## at that point. It is represented by ##\phi'## or ##\frac{d\phi}{dx}## and can be found using the methods mentioned in question 2.

5. Why is the derivative important in calculus?

The derivative is important in calculus because it allows us to analyze the behavior of functions and understand their rates of change. It is also used in many real-world applications, such as physics, economics, and engineering, to model and predict changes in variables over time.