Integral depending on a parameter

This note collects two standard “Leibniz-type” rules from Riemann integration.

Variable limits: $φ (x) = \int_{a (x)}^{b (x)} f (t) d t$

Assume $f$ is Riemann integrable on an interval $I$ and has an antiderivative $F$ on $I$ (e.g. it is continuous). Let $a, b : D \to I$ .

φ (x) = \int_{a (x)}^{b (x)} f (t) d t

is continuous on $D$ .

φ^{'} (x) = f (b (x)) b^{'} (x) - f (a (x)) a^{'} (x) .

Let $U \subset R$ be open and $f : U \times [a, b] \to R$ .

φ (x) = \int_{a}^{b} f (x, t) d t

is continuous on $U$ .

If additionally $\partial f / \partial x$ exists and is continuous on $U \times [a, b]$ , then $φ$ is differentiable and

φ^{'} (x) = \int_{a}^{b} \frac{\partial f}{\partial x} (x, t) d t .

Let $f : U \times I \to R$ be continuous, let $a, b : D \to I$ be differentiable with $a (x) < b (x)$ , and define

φ (x) = \int_{a (x)}^{b (x)} f (x, t) d t, x \in D .

Then $φ$ is differentiable on $D$ and (Leibniz)

φ^{'} (x) = f (x, b (x)) b^{'} (x) - f (x, a (x)) a^{'} (x) + \int_{a (x)}^{b (x)} \frac{\partial f}{\partial x} (x, t) d t .

Continuity only needs $f$ continuous (no $\partial_{x}$ ).
The three terms are exactly the three contributions: moving right limit, moving left limit, integrand moving.
Splitting into two steps: write $φ (x) = G (x, b (x)) - G (x, a (x))$ with $G (x, u) = \int_{c}^{u} f (x, t) d t$ , then apply the two rules above. The chain rule on $G$ gives the integral term; the $x$ -derivative of the lower anchor $c$ cancels because $c$ is constant.
Refinement: $f$ need not be continuous — it suffices that $f (x, \cdot) \in L^{1}$ for each $x$ , that $f (\cdot, t)$ is differentiable for each $t$ , and that there is an integrable (in $t$ ) majorant $g$ for both $| f (x, t) |$ and $| \partial_{x} f (x, t) |$ on $U_{0} \times [a, b]$ . This is dominated convergence, and it covers the pathological value of $f$ at a single $t$ -point that ruins continuity.