Question

find derivative ofcosx wrt x by first principle

Answer 1

$\bold{Answer :}$

Let us take,

y = f (x) = cosx

Then for h → 0,

f (x + h) = cos (x + h)

Now,

f (x + h) - f (x)

= cos (x + h) - cosx

= 2 sin {(x + h + x)/2} sin {(x - x - h)/2},

[ since cosC - cosD
= 2 sin {(C + D)/2} sin {(D - C)/2} ]

= - 2 sin (x + h/2) sin (h/2)

Now,

dy/dx

= lim (h → 0) {f (x + h) - f (x)}/h

= lim (h → 0) { - 2 sin (x + h/2) sin (h/2)}/h

= - lim (h → 0) {sin (h/2)}/(h/2)
× lim (h → 0) {sin (x + h/2)}

= - 1 (sinx), since lim (h → 0) (sinh)/h = 1

= - sinx

By 1st principle, we can conclude that,

d/dx (cosx) = - sinx

#$\bold{MarkAsBrainliest}$