Prove that

Question

d/dx (cos(x)) = -sin(x)?

Answer 1

According to fundamental theorem of the differentiation.

d/dx(cosx) = lim h--> 0 [(cos(x+h) - cos(x)) / h].

Using the identity cos(a+b) = cosacosb - sinasinb we get

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

= lim h--> 0 [(cosxcosh - sinxsinh - cos(x)) / h]

= lim h--> 0 [((cosx)(cosh - 1) / h) - (sinxsinh / h)]

= lim h--> 0 ((cosx)(cosh - 1) / h) - lim h--> 0 (sinxsinh / h)

= ((cosx)(cos0 - 1) / 0) - lim h--> 0 (sinxsinh / h)

= 0 - sinx lim h--> 0 (sinh / h)

=  - sinx (1)

=  - sinx