Partial Derivatives and the Fundamental Theorem of Calculus

Question

I am being asked to evaluate the 1st-order partial derivatives $-$ $f_{x}$(x,y) and $f_{y}(x,y)$ $-$ of the following multi-variable function: $f(x,y) = \int_{y}^{x} cos(-1t^2 + 3t -1) dt$. Any help is appreciated.

Answer 1

$$ f(x,y)=\int_y^x g(t) dt=G(t)|_y^x=G(x)-G(y) \text{ where } G'=g $$ so differentiating w.r.t $x$ on both sides of $f(x,y)=G(x)-G(y)$ gives $$ f_x(x,y)=(G(x)-G(y))_x$$ and by difference rule that is $$ f_x=(G(x))_x = g(x). $$

Answer 2

Apply Leibniz rule in each case.

For example

$$ f_x(x,y){}={}\dfrac{\partial}{\partial x}\int\limits^{x}_{y}\cos\left(-t^2+3t-1\right)\mathrm d t{}={}\cos\left(-x^2+3x-1\right)\dfrac{\mathrm d}{\mathrm d x}x{}={}\cos\left(-x^2+3x-1\right)\,. $$