partial derivative

[peacefreak77]

Find the partial derivatives of the function
f(x,y) =the integral from y to x of cos(-1 t^2 + 5 t + 2)dt

i think that the derivative of f(x,y) is the thing being integrated, but then my derivative has t's in it, and i don't know whether i'm supposed to actually try to integrate this, or somehow manipulate the derivative, or what...

i'm not sure how to find the partial derivative of an x,y function when given a derivative with t's in it.

i also can't figure out how to integrate this, u substitution doesn't work and i can't remember how to do that partial integration stuff, so i'm really stuck.

 

[galactus]

You do not have to actually integrate this function. It would be a booger to integrate anyway.


Think about the Second Fundamental Theorem of Calculus.

 

[peacefreak77]

okay, with the 2nd fundamental theorum (which i don't remember very well... i think we learned that two years ago) i can plug in x for t when i am taking the partial derivative with respect for x, but what do i do when i'm taking it with respect to y?

wow maybe i should go back and review this second fundamental theorum thing...

thanks

 

[peacefreak77]

could i do the the same thing except make it negative? (integrate backwards sorta)?

okay i just did that and it worked (my homework tells me if it's right or wrong). it worked because if you integrate backwords, from y to x, you can plug in y for t, and multiple it by -1? right?

thanks for helping me, i never would have remembered that. please just tell me if that's the reason why this worked and then i think i understand this problem.

 

[skeeter]

\(\displaystyle \L \frac{\partial}{\partial x}[\int_y^x f(t) dt] = f(x)\)

\(\displaystyle \L \frac{\partial}{\partial y}[\int_y^x f(t) dt] = -f(y)\)