Differentiation of a function in a domain

In summary, the problem asks for the derivatives of a function at an arbitrary point, but the problem fails to provide any information on how to find the derivatives in the given interval.
  • #1
paech
5
0

Homework Statement


Find the derivatives at an arbitrary point [itex]x[/itex] in the domain of the following functions [itex] f_i: D_i → ℝ[/itex], where for [itex]1 ≤ i ≤ 6[/itex] the domain [itex] D_i[/itex] is the maximal subset of [itex]ℝ[/itex] on which the mapping is defined - you don't have to determine the domains.

Homework Equations


a) [itex]f_1 (a) = (1-\sin3a)^5[/itex]

There are more functions for other parts of the question, but I just need help with understanding the problem.

The Attempt at a Solution


Ok, so I'm having trouble understanding what the question is really saying. From what I gather, it's saying that the domain it's listed is the set of real numbers in which the derivative will be defined. The domain it's listing has put me off, usually I would just substitute the arbitrary point [itex]x[/itex] into the function like so:[tex]f'(x) = (1-\sin3x)^5[/tex] and then apply the chain rule to find the derivative. I'm not sure what to do to find the derivative in the interval given.

Sorry if I haven't provided much of an attempt at a solution, I've looked over all my material and couldn't come up with anything.
 
Last edited:
Physics news on Phys.org
  • #2
paech said:

Homework Statement


Find the derivatives at an arbitrary point [itex]x[/itex] in the domain of the following functions [itex] f_i: D_i → ℝ[/itex], where for [itex]1 ≤ i ≤ 6[/itex] the domain [itex] D_i[/itex] is the maximal subset of [itex]ℝ[/itex] on which the mapping is defined - you don't have to determine the domains.

Homework Equations


a) [itex]f_1 (a) = (1-\sin3a)^5[/itex]

There are more functions for other parts of the question, but I just need help with understanding the problem.

The Attempt at a Solution


Ok, so I'm having trouble understanding what the question is really saying. From what I gather, it's saying that the domain it's listed is the set of real numbers in which the derivative will be defined. The domain it's listing has put me off, usually I would just substitute the arbitrary point [itex]x[/itex] into the function like so:[tex]f'(x) = (1-\sin3x)^5[/tex]
Was this a typo? [itex]f(x)= (1-\sin(3x))^5[/itex], not its derivative. And you do not substitute the value of x first and then differentiate! The value of a function is a constant and the derivative of a constant is always 0.

and then apply the chain rule to find the derivative. I'm not sure what to do to find the derivative in the interval given.

Sorry if I haven't provided much of an attempt at a solution, I've looked over all my material and couldn't come up with anything.
You are reading too much into this problem. All it is asking you to do is find the derivative- in its domain, which is nothing new. A function isn't defined outside its domain so this is just what you have been doing ever since you started differentiation. And the problem specifically says "you don't have to find the domains". Just differentiate!
 
  • #3
paech said:

Homework Statement


Find the derivatives at an arbitrary point [itex]x[/itex] in the domain of the following functions [itex] f_i: D_i → ℝ[/itex], where for [itex]1 ≤ i ≤ 6[/itex] the domain [itex] D_i[/itex] is the maximal subset of [itex]ℝ[/itex] on which the mapping is defined - you don't have to determine the domains.

Homework Equations


a) [itex]f_1 (a) = (1-\sin3a)^5[/itex]

There are more functions for other parts of the question, but I just need help with understanding the problem.

The Attempt at a Solution


Ok, so I'm having trouble understanding what the question is really saying. From what I gather, it's saying that the domain it's listed is the set of real numbers in which the derivative will be defined.
You made a slight error. Di is the domain of fi, not f'i, though the domains of the two could very well be the same.

The problem statement is just explicitly saying what you've likely been assuming all the time. When you define a function, you have to specify its domain, but it's common simply to say things like "Let f(x)=x+1" with the understanding that the domain is the set of all real numbers for which the function is defined. So like Halls said, the problem is really just asking you to find the derivatives like you normally do.
 
  • #4
Ah right, thanks. I had a feeling that I was misinterpreting the problem and that I just needed to differentiate it as I normally would.
 

What is differentiation of a function?

Differentiation of a function is a mathematical process that involves finding the rate at which the output of the function changes with respect to its inputs. In other words, it is the process of finding the slope of a function at a given point.

What is the domain of a function?

The domain of a function is the set of all possible input values for which the function is defined. It is the set of all x-values for which the function produces a valid output.

What is the difference between a derivative and a differential?

A derivative is the result of the differentiation process and represents the instantaneous rate of change of a function at a given point. A differential, on the other hand, is an infinitesimal change in the output of a function due to an infinitesimal change in the input.

What are the rules for differentiating a function?

There are several rules for differentiating a function, including the power rule, product rule, quotient rule, and chain rule. These rules allow us to find the derivative of a function by applying specific formulas depending on the form of the function.

How is differentiation used in real life?

Differentiation is widely used in various fields such as physics, engineering, economics, and statistics. It is used to analyze and model real-world phenomena, optimize processes, and make predictions. For example, in physics, differentiation is used to calculate the velocity and acceleration of an object at a given time. In economics, it is used to model the demand and supply curves of a market. In statistics, it is used to find the rate of change of a variable over time.

Similar threads

  • Calculus and Beyond Homework Help
Replies
9
Views
927
  • Calculus and Beyond Homework Help
Replies
3
Views
796
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
914
  • Calculus and Beyond Homework Help
Replies
10
Views
1K
  • Calculus and Beyond Homework Help
Replies
14
Views
910
  • Calculus and Beyond Homework Help
Replies
15
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
12
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
924
Back
Top