[{"data":1,"prerenderedAt":117},["ShallowReactive",2],{"word-differentiable":3},{"source":4,"data":5,"quality_score":115,"lastmod":116},"static_word_details",{"lemma":6,"phonetic":7,"pos":9,"definition_zh":11,"usage_points":12,"collocations":31,"examples_basic":73,"examples_advanced":83,"common_mistakes":93,"synonyms_nuance":102,"diff_pairs":111,"word_family":114},"differentiable",{"us":8,"uk":8},",difә'renʃiәbl, -ʃә-",[10],"形容词","在数学或计算中，指函数或曲线在某一点处可以求导，表示其变化率存在且连续。",[13,19,25],{"title":14,"content":15,"examples":16},"句型搭配","通常用作形容词，描述函数或变量的可微性，常与'be'动词或'become'等连用，表示状态或变化。",[17,18],"The function is differentiable at x=0.","It becomes differentiable after smoothing.",{"title":20,"content":21,"examples":22},"介词用法","常与'at'连用，指定可微的点或位置；在学术语境中，也可用'over'表示在某个区间内可微。",[23,24],"differentiable at a point","differentiable over the entire domain",{"title":26,"content":27,"examples":28},"语域说明","主要用于数学、物理、工程和计算机科学等学术或技术领域，日常对话中较少使用，属于专业术语。",[29,30],"In calculus, differentiable functions are key.","This model requires differentiable components.",[32,39,46,53,60,67],{"type":33,"items":34},"verb",[35,36,37,38],"be differentiable","become differentiable","remain differentiable","prove differentiable",{"type":40,"items":41},"adjective",[42,43,44,45],"continuously differentiable","piecewise differentiable","almost everywhere differentiable","smoothly differentiable",{"type":47,"items":48},"noun",[49,50,51,52],"differentiable function","differentiable manifold","differentiable curve","differentiable operator",{"type":54,"items":55},"adverb",[56,57,58,59],"everywhere differentiable","nowhere differentiable","almost differentiable","strictly differentiable",{"type":61,"items":62},"preposition",[63,64,65,66],"differentiable at","differentiable over","differentiable in","differentiable with respect to",{"type":68,"items":69},"phrase",[70,71,72],"differentiable almost everywhere","differentiable in the sense of","differentiable under certain conditions",[74,77,80],{"en":75,"zh":76},"In our math class, we learned that a smooth curve is differentiable at every point.","在数学课上，我们学到平滑曲线在每个点都是可微的。",{"en":78,"zh":79},"If the function has a sharp corner, it might not be differentiable there.","如果函数有尖锐的角，它在那里可能不可微。",{"en":81,"zh":82},"This simple equation is differentiable, so we can find its derivative easily.","这个简单的方程是可微的，所以我们可以轻松求出它的导数。",[84,87,90],{"en":85,"zh":86},"In machine learning, differentiable models allow for efficient gradient-based optimization during training.","在机器学习中，可微模型允许在训练期间进行高效的基于梯度的优化。",{"en":88,"zh":89},"The theorem states that a function is differentiable if it satisfies the Lipschitz condition in a neighborhood.","该定理指出，如果函数在邻域内满足利普希茨条件，那么它是可微的。",{"en":91,"zh":92},"For this financial model to work, we need all variables to be differentiable to apply calculus methods.","为了使这个金融模型有效，我们需要所有变量都是可微的，以便应用微积分方法。",[94,98],{"wrong":95,"right":96,"explanation":97},"This function is differentiable in every point.","This function is differentiable at every point.","错误在于使用了'in'，而正确介词是'at'，用于指定具体的点；'in'通常用于范围或区间，如'in the interval'。",{"wrong":99,"right":100,"explanation":101},"The curve is differentiable, so it must be continuous.","If a function is differentiable, then it is continuous.","错误在于因果表述不准确；正确用法应强调逻辑关系：可微性蕴含连续性，但反之不一定成立，需用条件句更严谨。",[103,107],{"word":104,"difference":105,"example":106},"smooth","'smooth'通常指函数无限次可微，具有更严格的连续性；而'differentiable'仅要求一次可导，可能在某些点不光滑。","A smooth function is always differentiable, but a differentiable function may not be smooth (e.g., with a kink).",{"word":108,"difference":109,"example":110},"derivable","'derivable'较少使用，更侧重于从某物推导出的含义；'differentiable'特指数学上的可微性，强调存在导数。","In logic, a conclusion might be derivable from premises, but in math, we say a function is differentiable at a point.",[112,113],"differentiable vs. derivable","differentiable vs. continuous",[],66,"2026-01-09T02:20:52.070Z",1781752496301]