🔗 真题
(2021·中山大学·高等数学I·期末考试·节选) 已知 $g(x)=\int_{0}^{1} f(tx) , dt$，求 $g’(x)$ .
解答
$$\begin{align}g’(x)&=\left( \frac{1}{x} \int_{0}^{x} f(tx) , dtx \right)’ \&= -\frac{1}{x^{2}}\int_{0}^{x} f(u) , du + \frac{f(x)}{x}\end{align}$$
方法论
变限积分求导
$$ (\int_{\phi_{1}(x)}^{\phi_{2}(x)} f(t) , dt)’ = f(\phi_{2}(x))\cdot \phi_{2}’(x) - f(\phi_{1}(x))\cdot \phi_{1}’(x) $$