Question

Find the derivative of f(x) = cos x at x = 0

Answer 1

We have,

$f\left(x\right)=cosx$

$\because f^{\prime }\left(a\right)=li{m}_{h\to 0}\frac{f(a+h)-f\left(a\right)}{h}$

$\because f^{\prime }\left(0\right)=li{m}_{h\to 0}\frac{f(0+h)-f\left(0\right)}{h}$

$=li{m}_{h\to 0}\frac{cos(0+h)-cos0}{h}$

$=li{m}_{h\to 0}\frac{cosh-cos0}{h}$

$=li{m}_{h\to 0}\frac{(1-\frac{h^2}{2!}+\frac{h^4}{4!}-...)-1}{h}$

$[\because cosx=1-\frac{x^2}{2!}+\frac{x^4}{4!}-....]$

$=li{m}_{h\to 0}\frac{h(-\frac{h}{2!}+\frac{h^3}{4!}-\frac{h^5}{6!})}{h}$

$=li{m}_{h\to 0}h(-\frac{h}{2!}+\frac{h^3}{4!}-\frac{h^5}{6!}-....)$

$=0$

$\therefore f^{\prime }\left(0\right)$