Find the derivative of cos x from first principle. - Vidyadip

Question

Find the derivative of cos x from first principle.

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Answer

Here f (x) = cos x
Then f (x + h) = cos (x + h)
We know that $f{\text{'}}(x) = \mathop {\lim }\limits_{h \to 0} \frac{{(x + h) - f(x)}}{h}$
$\Rightarrow f{\text{'}}(x) = \mathop {\lim }\limits_{h \to 0} \frac{{\cos (x + h) - \cos x}}{h}$
$ = \mathop {\lim }\limits_{h \to 0} \frac{{ - 2\sin \left( {\frac{{2x + h}}{2}} \right)\sin \left( {\frac{h}{2}} \right)}}{h}$
$ = \mathop {\lim }\limits_{h \to 0} - \sin \left( {\frac{{2x + h}}{2}} \right).\frac{{\sin \left( {\frac{h}{2}} \right)}}{{\frac{h}{2}}}$
= -sin x

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