Changing V(x) to V(t): Chain Rule Application?

In summary, the conversation discusses changing from velocity in terms of position to velocity in terms of time. The solution involves using the chain rule and integrating both sides to find t as a function of x, then solving for x as a function of t. This can also be achieved by finding the antiderivative. The conversation concludes with thanks and confirmation that the problem was solved.
  • #1
StephenSF8
3
0
I have a function for velocity, V in terms of position, x. The equation is of the form V(x) = a*x2+b*x+c. Initial conditions are x=0, t=0.

How do I change from V(x) to V(t)? It seems this would be an application of the chain rule, dy/dx = dy/du * du/dx, but I'm struggling to adapt it to this situation. Am I way off base?

Thanks!
 
Physics news on Phys.org
  • #2
Assuming t means time, then V=dx/dt. So dt = dx/(a*x2+b*x+c). Integrate both sides to get t as a function of x. Solve for x as a function of t. Then take the derivative to get V. Good luck!
 
  • #3
Velocity is the result of differentiating a position equation. To reverse that you have two options, (the same pretty much): Integrate or find the Antiderivative.

Enjoy.
 
  • #4
mathman said:
Assuming t means time

You assume correct.

mathman said:
Integrate both sides to get t as a function of x. Solve for x as a function of t.

This is what I was missing. I was able to perform the integration and solve for x and all is well. Thanks for the help!
 

Related to Changing V(x) to V(t): Chain Rule Application?

1. How do you apply the chain rule to change V(x) to V(t)?

The chain rule is a mathematical rule used to find the derivative of a composite function. In this case, we are changing the independent variable from x to t. To apply the chain rule, we need to take the derivative of the outer function (V) with respect to the inner function (x) multiplied by the derivative of the inner function with respect to the new variable (t). This can be written as dV/dt = (dV/dx)*(dx/dt).

2. Why is it necessary to change V(x) to V(t)?

In some scientific experiments and calculations, it may be more useful to express a variable with respect to a different independent variable. For example, in physics, it may be more convenient to express velocity (V) with respect to time (t) instead of position (x). Changing V(x) to V(t) allows us to better understand the relationship between the variables and make calculations or predictions based on the new representation.

3. What are the steps involved in changing V(x) to V(t)?

The steps involved in changing V(x) to V(t) include identifying the outer and inner functions, taking the derivative of the outer function with respect to the inner function, and then multiplying it by the derivative of the inner function with respect to the new variable. It is important to remember to use the chain rule and to keep track of the variables in each step.

4. Can the chain rule be applied to any function?

Yes, the chain rule can be applied to any function that is a composition of two or more functions. However, it is important to note that the chain rule may become more complex when dealing with multiple variables or functions with higher orders of derivatives.

5. Are there any real-world applications of changing V(x) to V(t) using the chain rule?

Yes, there are many real-world applications of the chain rule and changing variables. For example, in physics, it can be used to calculate the acceleration of an object in motion by changing the velocity function from V(x) to V(t). In economics, it can be used to analyze the relationship between two variables, such as price and demand, by changing the independent variable. These are just a few examples, but the chain rule is a fundamental tool used in many fields of science and mathematics.

Similar threads

I Second derivative, chain rules and order of operations
    • Calculus
Replies
6
Views
2K
I Exploring the Derivative of y(x,t) in Quantum Mechanics
    • Calculus
Replies
1
Views
933
B Integration by Parts, an introduction I get confused with
    • Calculus
Replies
2
Views
1K
I Is My Proof of the Chain Rule Correct?
    • Calculus
Replies
2
Views
1K
I Reasoning behind Infinitesimal multiplication
    • Calculus
Replies
22
Views
2K
MHB Calculating Derivative of Integral w/ Chain Rule
    • Calculus
Replies
9
Views
2K
Understanding the Chain Rule: (df/dx) + (df/dy)* (dy/dx)
    • Calculus
Replies
1
Views
1K
Insights How to Solve Second-Order Partial Derivatives
    • Calculus
Replies
3
Views
3K
I Integration of ##e^{-x^2}## with respect to ##x##
    • Calculus
Replies
4
Views
2K
The below solution seems to assume that 1/0 = 0
    • Calculus and Beyond Homework Help
Replies
7
Views
894

Hot Threads

Recent Insights

Back
Top