I am unsure where to begin with finding this partial derivative.
f(x,y)=integral from y to x of cos(t^2)dt
$\displaystyle f(x, y) = \int_y^x \cos (t^2) \, dt$.
Using the Fundamental Theorem of Calculus and noting that y is treated as a constant when you differentiate with respect to x:
$\displaystyle \frac{\partial f}{\partial x} = \cos(x^2)$.
Now note that $\displaystyle f(x, y) = \int_y^x \cos (t^2) \, dt = - \int_x^y \cos (t^2) \, dt $ and use the Fundamental Theorem of Calculus again (note that x is treated as a constant when you differentiate with respect to y):
$\displaystyle \frac{\partial f}{\partial y} = - \cos(y^2)$.
Depends on what partial derivative you want
$\displaystyle f(x,y) = \int_y^x \cos t^2\, dt = -\int_x^y \cos t^2 dt$
so using the 2nd fundemental theorem of Calculus
$\displaystyle \frac{\partial f }{\partial x } = \frac{\partial }{\partial x } \int_y^x \cos t^2 dt = \cos x^2$
$\displaystyle \frac{\partial f }{\partial y } = - \frac{\partial }{\partial y } \int_x^y \cos t^2 dt = - \cos y^2$