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Question Number 148783 by 0731619 last updated on 31/Jul/21

Commented by hknkrc46 last updated on 31/Jul/21

$$\left(1\right)\underset{\mathit{h}\to 0}{\lim }\frac{\mathit{f}(\mathit{x}+\mathit{h})-\mathit{f}\left(\mathit{x}\right)}{\mathit{h}}$$$$=\underset{\mathit{h}\to 0}{\lim }\frac{-\cos (\mathit{x}+\mathit{h})+\cos \mathit{x}}{\mathit{h}}$$$$=\underset{\mathit{h}\to 0}{\lim }\frac{\cos \mathit{x}-\cos \mathit{x}\cdot \cos \mathit{h}+\sin \mathit{x}\cdot \sin \mathit{h}}{\mathit{h}}$$$$=\underset{\mathit{h}\to 0}{\lim }\frac{\cos \mathit{x}(1-\cos \mathit{h})}{\mathit{h}}+\underset{\mathit{h}\to 0}{\lim }\frac{\sin \mathit{x}\cdot \sin \mathit{h}}{\mathit{h}}$$$$=\cos \mathit{x}\cdot \underset{\mathit{h}\to 0}{\lim }\frac{1-\cos \mathit{h}}{\mathit{h}}+\sin \mathit{x}\cdot \underset{\mathit{h}\to 0}{\lim }\frac{\sin \mathit{h}}{\mathit{h}}$$$$=\cos \mathit{x}\cdot 0+\sin \mathit{x}\cdot 1=\sin \mathit{x}$$$$★\frac{\mathit{d}(-\cos \mathit{x})}{\mathit{d}\mathit{x}}=\sin \mathit{x}\iff \int \sin \mathit{x}\mathit{d}\mathit{x}=-\cos \mathit{x}+\mathit{c}$$$$$$

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