I'm having difficulties grasping why Leibniz rule the way it is:
$$
\frac{d}{dx} \int^{b(x)}_{a(x)}f(x,t)dt = f(x,b(x))\frac{db}{dx} – f(x, a(x))\frac{da}{dx} + \int^{b(x)}_{a(x)}\frac{\partial}{\partial x}f(x,t)dt
$$
Specifically, I don't understand how the last integral term comes to be? i.e.
$$
\int^{b(x)}_{a(x)}\frac{\partial}{\partial x}f(x,t)dt
$$
I think I understand the source of the first 2 terms:
Let $\frac{dF(x)}{dx} = f(x)$, then
$$
\int^{b(x)}_{a(x)}f(x,t)dt = F(x,b(x)) – F(x,a(x))
$$
differentiating:
$$
\frac{d}{dx}[F(x,b(x)) – F(x, a(x))] = f(x, b(x))\frac{db}{dx} – f(x, a(x))\frac{da}{dx}
$$
$\frac{db}{dx}$ and $\frac{da}{dx}$ come from Chain Rule
Where does
$$
\int^{b(x)}_{a(x)}\frac{\partial}{\partial x}f(x,t)dt
$$
come from, what am I missing?
Best Answer
Note that $f$ and $F$ are bivariate functions and that $F$ corresponds to the antiderivative of $f$ with respect to the second argument only, i.e. $f(x,y) = \partial_yF(x,y)$. When dealing with several variables, it is often easier to work with differentials. Here, one has : $$ \mathrm{d}F(x,y) = \frac{\partial F}{\partial x}\mathrm{d}x + \frac{\partial F}{\partial y}\mathrm{d}y = \partial_xF(x,y)\mathrm{d}x + f(x,y)\mathrm{d}y, $$ hence $$ \frac{\mathrm{d}}{\mathrm{d}x} \left(F(x,b(x))-F(x,a(x))\right) = \left[\partial_xF(x,y) + f(x,y)\frac{\mathrm{d}y}{\mathrm{d}x}\right]_{y=a(x)}^{y=b(x)} = \partial_xF(x,b(x)) - \partial_xF(x,a(x)) + f(x,b(x))b'(x) - f(x,a(x))a'(x), $$ where $$ \partial_x\left(F(x,b(x))-F(x,a(x))\right) = \partial_x \int_{a(x)}^{b(x)} f(x,y) \,\mathrm{d}y = \int_{a(x)}^{b(x)} \partial_xf(x,y) \,\mathrm{d}y. $$ In conclusion, this last term comes from the partial derivative of $f$ with respect to its first argument. It is to be noted that this partial derivative commutes with the integral because the $x$-dependent bounds of the integral come into play through its second argument only, while the total derivative doesn't, because it takes into account the contributions coming from both arguments (hence the said Leibniz integral rule).