Derivative of sinx Since sin(x + y) = sinxcosy + cosxsiny we have d

Derivative of sinx Since sin(x + y) = sinxcosy + cosxsiny we have d

Derivative of sin x
Since sin(x + y) = sin x cos y + cos x sin y we have
sin(x + h) − sin x
d
(sin x) = lim
dx
h
h→0
sin x cos h + cos x sin h − sin x
= lim
h
h→0
cos x sin h sin x cos h − sin x
= lim
+
h
h
h→0
cos h − 1
sin h
+ sin x · lim
= cos x · lim
h
h→0
h→0 h
1
sin h
tan h
h
≤
1
2
≤
sin h ≤ 12 h ≤ 12 tan h
⇒1≤
h
1
≤
sin h
cos h
⇒ cos h ≤
sin h
≤1
h
sin h
=1
h→0 h
lim cos h = 1 ⇒ lim
h→0
cos h − 1
cos2 h − 1
= lim
h→0
h→0 h(cos h + 1)
h
lim
− sin2 h
h→0 h(cos h + 1)
= lim
= cos x · 1 + sin x · 0
= lim
h→0
= cos x
sin h − sin h
·
h
cos h + 1
=1·
Peter Jipsen — Math Poster 2011 — math.chapman.edu
−0
=0
1+1