导数：ecosxe^{\cos x}ecosx Derive from first principles 专题 General / 综合 难度 L4 来源 QuantQuestion 题目详情 Derive, from first principles, the derivative of g(x)=ecos(x)g(x) = e^{\cos (x)}g(x)=ecos(x) 解析 由定义 g′(x)=limh→0ecos(x+h)−ecosxh.g'(x)=\lim_{h\to 0}\frac{e^{\cos(x+h)}-e^{\cos x}}{h}.g′(x)=h→0limhecos(x+h)−ecosx. 将差分因式分解： ecos(x+h)−ecosxh=ecosx⋅ecos(x+h)−cosx−1h.\frac{e^{\cos(x+h)}-e^{\cos x}}{h}=e^{\cos x}\cdot\frac{e^{\cos(x+h)-\cos x}-1}{h}.hecos(x+h)−ecosx=ecosx⋅hecos(x+h)−cosx−1. 又 cos(x+h)−cosx=−sinx h+o(h)\cos(x+h)-\cos x=-\sin x\,h+o(h)cos(x+h)−cosx=−sinxh+o(h)，并且 limu→0eu−1u=1\lim_{u\to 0}\frac{e^u-1}{u}=1limu→0ueu−1=1，因此 g′(x)=−sinx ecosx.\boxed{g'(x)=-\sin x\,e^{\cos x}}.g′(x)=−sinxecosx.