Questions

$$\frac{d}{dx}\,\int_{0}^{x^{2}} \cos(t)\,dt =\ ?$$

$$\frac{d}{dx}\,\int_{g(x)}^{f(x)} h(t)\,dt =\ ?$$

$$\frac{\partial}{\partial y}\,\int_{x^{2}y}^{\,y\sin x}\arctan(t)\,dt =\ ?$$

$$\frac{\partial}{\partial x}\,\int_{\,f(x,y)}^{\,g(x,y)} h(t)\,dt =\ ?$$

$$\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_n}{\int }_{g(x_1,\dots ,x_n)}^{f(x_1,\dots ,x_n)}h(t)dt=$$

Solutions

1)
Here is how I like to think of the fundamental theorem of calculus:

$$\int_0^{g(x)} f(t)\,dt=F(g(x))-F(0)$$

Notice that $F(g(x))$ is a function, while $F(0)$ is a constant.
When we take the derivative,

$$\frac{d}{dx}\int_0^{g(x)} f(t)\,dt=\frac{d}{dx}\big(F(g(x))-F(0)\big) =F'(g(x))\cdot g'(x)-0=f(g(x))\cdot g'(x).$$

With this logic,

$$\frac{d}{dx}\int_{0}^{x^{2}}\cos(t)\,dt =2x\cos(x^{2}).$$

2)

$$\frac{d}{dx}\int_{g(x)}^{f(x)} h(t)\,dt =\frac{d}{dx}\big[H(f(x))-H(g(x))\big] =H'(f(x))f'(x)-H'(g(x))g'(x) =h(f(x))f'(x)-h(g(x))g'(x).$$

3)
Similar process as other fundamental theorem of calculus problems, except with partial derivatives.

$$\frac{\partial}{\partial y}\int_{x^{2}y}^{\,y\sin x}\arctan(t)\,dt =\arctan(y\sin x)\cdot \sin x\;-\;\arctan(x^{2}y)\cdot x^{2}.$$

4)

$$\frac{\partial}{\partial x}\int_{\,f(x,y)}^{\,g(x,y)} h(t)\,dt =\frac{\partial}{\partial x}\big[H(g(x,y))-H(f(x,y))\big] =H'(g)\,\frac{\partial g}{\partial x}-H'(f)\,\frac{\partial f}{\partial x} =h(g)\,\frac{\partial g}{\partial x}-h(f)\,\frac{\partial f}{\partial x}.$$

5)

$$\frac{\partial}{\partial x_n}\int_{\,g(x_1,\dots,x_n)}^{\,f(x_1,\dots,x_n)} h(t)\,dt =h\big(f(\mathbf{x})\big)\,\frac{\partial f}{\partial x_n} -h\big(g(\mathbf{x})\big)\,\frac{\partial g}{\partial x_n}.$$