Joel Kammet

Posted on Saturday, 08 November, 2003 - 04:33 am:

I am to find the first partial derivatives of

f(x,y)=ň_x^y cos(t²) dt

are they simply

f_x=-2x cos(x²)

and

f_y=2y cos(y²)

If not, please explain how to do this.

Joel Kammet

Posted on Saturday, 08 November, 2003 - 08:00 am:

That was supposed to be:
f(x,y) = LaTeX Image

cos(t² )dt

I'm pretty confused by this.
Isn't this a composite function that calls for use of the chain rule?

I'm thinking that I have g(t) = t² and the integral is f(g(t)) which gets evaluated at t=y and t=x, thus becoming f(g(x)) and f(g(y)), so cos(y² ) - cos(x² ) is df/dg and I still have to differentiate that wrt x to get LaTeX Image

x and wrt y to get LaTeX Image

y.

So,
LaTeX Image

x = df/dg* LaTeX Image

x
and
LaTeX Image

y = df/dg * LaTeX Image

x

Then,
LaTeX Image

x =

x[cos(y² ) - cos(x² )] = -2xcos(x² )
and
LaTeX Image

y =

y[cos(y² ) - cos(x² )] = 2ycos(y² )

But I'm not very confident about this. Please tell me if it's all nonsense.

Andre Rzym

Posted on Saturday, 08 November, 2003 - 08:42 am:

It's not correct, I'm afraid. Try writing śf/śx (partial derivative) as a limit as DŽ 0, and substitute the integral into the limit expression. You should see that 'most of' the integral cancels (you have the difference of two integrals which just differ by their limits), and you are then interested in evaluating an integral from x to x+D.

I'll post more later if that makes no sense.

Andre

Kerwin Hui

Posted on Saturday, 08 November, 2003 - 08:44 am:

There should not be a factor of 2x (or 2y respectively). If you write the integral in the form
LaTeX Image

for some fixed point p, the question reduces to applying the fundamental theorem of calculus, i.e.
LaTeX Image

and similar.

Kerwin

Joel Kammet

Posted on Saturday, 08 November, 2003 - 07:00 pm:

OK. I really needed to go back & think about the fundamental theorem for a while. The "difference" aspect is clear. I was just getting confused about which function I was dealing with.
i.e. LaTeX Image

vs.

and ending up with a mishmash of the two.

I've had better moments.

Thanks, both of you, for your help.