What are the first partial derivatives of a definite integral function?

Click For Summary
The discussion focuses on finding the first partial derivatives of the function f(x,y) defined as the definite integral from x to y of cos(t^2) dt. The initial attempt suggests that the partial derivative with respect to x is cos(x^2) and raises a question about the derivative with respect to y, contemplating whether it should be negative. Participants confirm that the partial derivative with respect to y should indeed be -cos(y^2), in accordance with the Leibniz rule for differentiating under the integral sign. The conversation emphasizes the application of the Leibniz rule to correctly determine the derivatives. Understanding these derivatives is crucial for further analysis of the function's behavior.
icosane
Messages
48
Reaction score
0

Homework Statement



Find the first partial derivatives of the function.

f(x,y) = definite integral (limits of integration x to y) cos(t^2) dt


The Attempt at a Solution



Is the partial derivative with respect to x just cos(x^2), and for y, cos(y^2) ? Or should the partial derivative with respect to y be negative?
 
Physics news on Phys.org
Sure. -cos(y)^2 for the y derivative. That's what the Leibniz rule says, right?
 

Thread 'Distance between a Clock's hands when the distance is increasing most rapidly'

Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?
View full post »

Similar threads

Determining domain for C^1 function
  • · Replies 4 ·
Replies
4
Views
1K
Find f(x,y) given partial derivative and initial condition
  • · Replies 6 ·
Replies
6
Views
2K
Discontinuous partial derivatives example
  • · Replies 2 ·
Replies
2
Views
1K
Partial Derivative Simplification
  • · Replies 3 ·
Replies
3
Views
2K
Multivariable calculus problem involving partial derivatives along a surface
  • · Replies 9 ·
Replies
9
Views
2K
Calculate this Integral around the Circular Path using Green's Theorem
  • · Replies 3 ·
Replies
3
Views
2K
Stokes' Theorem for a Circle in a Plane
  • · Replies 7 ·
Replies
7
Views
2K
Directional derivatives vs Partial derivatives
  • · Replies 7 ·
Replies
7
Views
2K
How should I show that this system has no limit cycle?
  • · Replies 2 ·
Replies
2
Views
1K
Multivariable calculus proof involving the partial derivatives of an expression
  • · Replies 4 ·
Replies
4
Views
1K