How would one reasonable differentiate the integral of the form $\int_{f(x)}^{g(x)} h(x) dx$, provided $h$ is continuous? Does this even make reasonable sense? The main point is that the integrand is also a function of $x$.
2026-04-30 07:56:25.1777535785
Differentiating an integral with dependent limits
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$$\int_{f(x)}^{g(x)} h(x) dx = \int_{0}^{g(x)} h(x) dx -\int_{0}^{f(x)} h(x) dx$$
Apply the Fundamental Theorem of Calculus to get
$$ \frac {d}{dx} \int_{f(x)}^{g(x)} h(x) dx = h(g(x))g'(x) - h(f(x))f'(x) $$