# multivariate derivative of integrals

Is it true that \begin{align} \frac{d^2}{dy_1dy_2}\left(\int_0^{f(y_2)}\int_0^{f(y_1)}g(x_1, x_2)dx_1dx_2\right)= f'(y_2)f'(y_1)g(y_1, y_2) \end{align}?

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No it's $f'(y_2)f'(y_1)g(f(y_1),f(y_2))$ as a simple example shows $$\frac{d}{dx}\int_0^{x^2}\cos(y)dy=\frac{d}{dx}\left[ \sin(y) \right]_0^{x^2}=\frac{d}{dx}\sin(x^2)=2x\cos(x^2)$$