Calculate $ \frac{\partial f}{\partial x} (x,y) $ Calculate $ \frac{\partial f}{\partial x} (x,y) $ of
$$ f(x,y) = \int_{x^2}^{y^2} e^{-t^2}\, dt$$

Let us call $E=E(t)$ a fixed antiderivative of $e^{-t^2}$. Then $$f(x,y)=E(y^2)-E(x^2).$$ Now, $$\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} \left( E(y^2)-E(x^2) \right) = -2x E'(x^2)=-2x e^{-x^4}.$$
