Please help me solve it use the defndferentiabilty at a poin

use the defndferentiabilty at a point to find the derivative of f(x)= cos(x) 2) Gvn the functionl equation g in differentiable at =0. b) Find the imct expression fr g o)

Solution

note : only one question allowesd per submission

1.

Using the definition of the derivative,

lim [ cos(x + h) - cos(x) ] / h
h -> 0

lim [ cos(x)cos(h) - sin(x)sin(h) - cos(x) ] / h triigonometric formula
h -> 0

lim [ cos(x)cos(h) - cos(x) - sin(x)sin(h) ] / h rearranging
h -> 0

lim [ cos(x) [ cos(h) - 1 ] - sin(x)sin(h) ] / h factoring
h -> 0

lim [ cos(x) [ cos(h) - 1 ]/h - [sin(x)sin(h)]/h rearranging and writing seperately
h -> 0

lim [ cos(x) [cos(h) - 1]/h ] - lim sin(x)sin(h)/h
h -> 0 . . . . . . . . . . . . . . . . . . h -> 0

Factor cos(x) from the first limit, and sin(x) from the second limit.

cos(x) lim [ cos(h) - 1 ]/h - sin(x) lim sin(h)/h
. . . . . . h -> 0 . . . . . . . . . . . . . . . . . h -> 0

lim [ cos(h) - 1 ]/h = 0, and
h -> 0
lim sin(h)/h = 1
h -> 0

The above then becomes

cos(x) (0) - sin(x)(1)
0 - sin(x)

-sin(x)