一条曲线的方程为 $\ln(x+y)=3x^2y$。

求该曲线在点 $(1,0)$ 处的斜率。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

两边隐式求导后代入给定点，解出导数。

Differentiating implicitly,

At $(1,0)$, let the gradient be $m$. Then

giving $\boxed{m=\frac12}$.

也可先指数化再求导。在给定点指数项等于一，所得斜率方程相同。

Exponentiating the curve equation gives

Differentiating,

At $(1,0)$,

so the gradient is $\boxed{m=\frac12}$.