Calculate $ \frac{\partial f}{\partial x} (x,y) $ of $$ f(x,y) = \int_{x^2}^{y^2} e^{-t^2}\, dt$$
2026-04-03 02:34:19.1775183659
Calculate $ \frac{\partial f}{\partial x} (x,y) $
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Let us call $E=E(t)$ a fixed antiderivative of $e^{-t^2}$. Then $$f(x,y)=E(y^2)-E(x^2).$$ Now, $$\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} \left( E(y^2)-E(x^2) \right) = -2x E'(x^2)=-2x e^{-x^4}.$$