---
tag: 0AR06W1E1B
kind: statement
role: theorem
name: Differentiation under the integral sign
aliases: [leibniz-integral-rule]
msc: [26B15]
uses: [05AXG3QRKR, 0S9EK5MQPE, 0EAAS2EHDR]
sources:
  - url: https://en.wikipedia.org/wiki/Leibniz_integral_rule
    locator: 'General form: differentiation under the integral sign'
  - url: https://en.wikipedia.org/wiki/Leibniz_integral_rule
    locator: Measure theory statement
same_as:
  wikidata: Q2996637
  wikipedia: Leibniz integral rule
---

## Statement

Let $f \colon [a, b] \times [t_0, t_1] \to \mathbb{R}$ be continuous, and suppose that its [[partial derivative|05AXG3QRKR]] $\partial f / \partial t$ exists and is [[continuous|0S9EK5MQPE]] on $[a, b] \times [t_0, t_1]$.
Then $F(t) = \int_a^b f(x, t) \, dx$ is continuously differentiable on $[t_0, t_1]$, and
$$F'(t) = \int_a^b \frac{\partial f}{\partial t}(x, t) \, dx.$$
If, in addition, $\alpha, \beta \colon [t_0, t_1] \to [a, b]$ are continuously differentiable, then
$$\frac{d}{dt} \int_{\alpha(t)}^{\beta(t)} f(x, t) \, dx = \int_{\alpha(t)}^{\beta(t)} \frac{\partial f}{\partial t}(x, t) \, dx + f(\beta(t), t) \, \beta'(t) - f(\alpha(t), t) \, \alpha'(t).$$

## Remarks

The second formula follows from the first, the fundamental theorem of calculus and the [[chain rule|0EAAS2EHDR]], applied to $G(u, v, t) = \int_u^v f(x, t) \, dx$.
A measure theory version: let $f(\cdot, t)$ be integrable for each $t$ in an open set, let $\partial f / \partial t$ exist for almost all $x$, and let $|\partial f / \partial t (x, t)| \le g(x)$ for an integrable $g$. Then the same rule holds. Its proof uses the dominated convergence theorem.
Example: let $F(t) = \int_0^1 \frac{x^t - 1}{\ln x} \, dx$ for $t \ge 0$. For $t > 0$ the measure version, with $g = 1$, gives $F'(t) = \int_0^1 x^t \, dx = \frac{1}{t + 1}$. Since $F$ is continuous with $F(0) = 0$, $F(t) = \ln(1 + t)$.

## Explanations

### L1 human

If the integrand depends smoothly on a parameter, you can differentiate the integral by differentiating inside it.
When the limits of integration also move, the values of the integrand at the moving limits add extra terms.