题目
设 $\left\{\begin{matrix}x = \arctan t,\\ y = 3t + t^{3},\end{matrix}\right.$ 则 $\frac{d^{2}y}{dx^{2}}|_{t=1} = ?$
解析
本题就是复合函数求导。
$$
\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}.
$$
$$
\frac{dy}{dt} =
$$
$$
3+3t^{2}.
$$
$$
\frac{dx}{dt} =
$$
$$
\frac{1}{1+t^{2}}.
$$
$$
y^{‘} = \frac{dy}{dx} =
$$
$$
(3+3t^{2})(1+t^{2}) =
$$
$$
3+3t^{2} + 3t^{2} + 3t^{4} =
$$
$$
3+6t^{2}+3t^{4}.
$$
$$
\frac{dy^{‘}}{dx} =\frac{\frac{dy^{‘}}{dt}}{\frac{dx}{dt}}.
$$
$$
\frac{dy^{‘}}{dt} =
$$
$$
12t + 12t^{3}.
$$
$$
\frac{dy^{‘}}{dx} =
$$
$$
(12t + 12t^{3})(1+t^{2}).
$$
当 $t=1$ 时，有：
$$
(12t + 12t^{3})(1+t^{2}) =
$$
$$
(12+12)(1+1) =
$$
$$
24 \times 2 = 48.
$$
综上可知，正确答案为 $48$.
EOF