Theorem

For an integral of the form ∫a(z)b(z)f(x,z) dx\int_{a(z)}^{b(z)} f(x,z)\ dx,

∂∂z∫a(z)b(z)f(x,z) dx=∫a(z)b(z)∂f∂z dx+f(b(z),z)∂b∂z−f(a(z),z)∂a∂z\frac{\partial}{\partial z}\int_{a(z)}^{b(z)}f(x,z)\ dx = \int_{a(z)}^{b(z)}\frac{\partial f}{\partial z}\ dx+f(b(z),z)\frac{\partial b}{\partial z}-f(a(z),z)\frac{\partial a}{\partial z}

References

  1. https://mathworld.wolfram.com/LeibnizIntegralRule.html
  2. https://en.wikipedia.org/wiki/Leibniz_integral_rule
  3. https://zackyzz.github.io/feynman.html
  4. https://tracingcurves.wordpress.com/2023/04/02/leibniz-rule-differentiate-under-the-integral-sign-general-case/