Question

How do you differentiate $cosx?$

Answer 1

Differentiate cos x using the first principle of derivatives :

$\begin{array}{rcl}\frac{d}{dx}(\cos x)& =& \underset{h\to 0}{\lim }\frac{\left[\cos \right(x+h)–\cos x]}{h}\left[\underset{\mathbf{h}\mathbf{\to }\mathbf{0}}{\mathbf{l}\mathbf{i}\mathbf{m}}\mathbf{}\frac{\mathbf{\left[}\mathbf{f}\mathbf{\right(}\mathbf{x}\mathbf{}\mathbf{+}\mathbf{}\mathbf{h}\mathbf{)}\mathbf{}\mathbf{–}\mathbf{}\mathbf{f}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right]}}{\mathbf{h}}\mathbf{}\right]\\ & =& \underset{h\to 0}{\lim }\frac{[\cos x\cos h–\sin x\sin h–\cos x]}{h}\\ & =& \underset{h\to 0}{\lim }\frac{\cos x\left[\right(\cos h–1\left)\right]}{h}-\frac{\sin x(\sin h)}{h}\\ & =& 0–\sin x\left[\underset{\mathbf{x}\mathbf{\to }\mathbf{0}}{\mathbf{l}\mathbf{i}\mathbf{m}}\mathbf{}\mathbf{}\frac{\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{x}}{\mathbf{x}}\mathbf{}\mathbf{=}\mathbf{1}\right]\\ & =& -\sin x\end{array}$

Hence ,the differentiation of $\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathit{x}$ is $\mathbf{-}\mathit{s}\mathit{i}\mathit{n}\mathit{x}\mathbf{}\mathbf{.}$