Leibniz integral rule and Fundamental thm of Calculs(differences) What is difference between Leibniz integral rule and F.T.C.?
Thank you
 A: 
Leibniz Integral Rule (Differentiation under the integral sign):
Let $f(x, t)$ be a function of $x$ and $t$ such that both $f(x, t)$ and its partial derivative $\frac{\partial f}{\partial x}$ are continuous in $t$ and $x$ in some region of the $(x, t)$-plane, including $a(x) ≤ t ≤ b(x)$, and $ x_0 ≤ x ≤ x_1$. Also suppose that the functions $a(x)$ and $b(x)$ are both continuous and both have continuous derivatives for $x_0 ≤ x ≤ x_1$. Then, for $x_0 ≤ x ≤ x_1$,
$$\frac{d}{dx}(\int_{a(x)}^{b(x)} f(x,t) dt)=\int_{a(x)}^{b(x)} \frac{\partial }{\partial x}f(x,t) dt +f( x, b(x)) \frac{db}{dx}-f( x, a(x)) \frac{da}{dx}$$



Fundamental Theorem of Calculus:
Suppose that the function $F(x)$ is differentiable everywhere on $[a, b]$ and if $f$ is integrable on $[a, b]$. Also suppose that $F$ is an anti-derivative for $f$, i.e., if $F'=f$, then
$$\int_a^b f(x) dx = F(b)-F(a)$$

If $a(x)$ and $b(x)$ are constants (i.e., $a\neq a(x)$ and $b \neq b(x)$), then $\frac{da}{dx}=0$ and $\frac{db}{dx}=0$ and hence the first theorem becomes,$$\frac{d}{dx}(\int_{a}^{b} f(x,t) dt)=\int_{a}^{b} \frac{\partial }{\partial x}f(x,t) dt $$
Now in additional if $f(x,t)=f(x)$, then the above equation becomes the fundamental theorem of calculus.
So we say that the fundamental theorem of calculus is just the particular case of the Leibniz integral rule.
