Question

Let f(x, y) = integral(1, x) of P(t) dt
+ integral(1, y) Q(t) dt. Find df/dx, df/dy

Answer 1

So
\[f(x,y) = \int\limits\limits_{1}^{x} P(t) dt +\int\limits\limits_{1}^{y} Q(t) dt\]
Use the First Fundamental Theorem of Calculus:
if
\[g(x)=\int\limits_{a}^{x}h(t)dt\]
then
\[\frac{d \ g(x)}{d \ x}=h(x)\]

So in this particular problem we get:
\[\frac{df}{dx}=P(x)+0\]
the zero is because the second integral is a function of y and dy/dx is 0.
\[\frac{df}{dy}=0+Q(y)\]