Based on the Fundamental theorem of calculus . Is that correct?

Question

Based on the Fundamental theorem of calculus? Is that correct?

$$\dfrac{\partial}{\partial y}\int_{a}^{b}dx f(x,y)=f(b,y)-f(a,y)$$

Thanks

Answer 1

$\text{No, that is not correct because you are integrating with respect to } x\; \text{ and differentiating with respect to} \;\;y.\text{What is true is that}\\\displaystyle\dfrac{\partial}{\partial y}\int_a^b f(x,y)dx= \int_a^b \frac{\partial f(x,y)}{\partial y}dx\\\text{The integral of }\dfrac{\partial f}{\partial y}\text{ with respect to } y\text{ would be}\\f(x,b) - f(x,a)\tag*{}$