Theorems

Theorem: Derivative of Sine Function: D x ( sinx )=cos x

Proof:

Let f be the sine function so that f(x) = sinx

From the definition of derivative,

f’(x) = lim△ x→0

f ( x+△ x )−f (x )

△ x

= lim△ x→0

sin ( x+△ x )−sinx

△ x

f’(x) = lim△ x→0

sinxcos (△ x )+cosxsinx (△ x )−sinx

△ x

= lim△ x→0

sinx [cos (△ x )−1 ]△ x

+lim△ x→0

cosxsin (△ x )

△ x

= − lim

△ x→01−cosx (△ x )

△ x ( lim△ x→0sinx )+( lim△ x→0

cosx )lim△ x→0

sin (△ x )

△ x

From the theorems: lim△ x→0

1−cos (△ x)

△ x = 0

lim△ x→0

sin (△ x)

△ x=1

Substituting from these equations:

f’(x) = −0 · sinx+cosx ·1

f’(x) = cosx

Theorem: Derivative of Cosine Function: D x (cosx )=−sin x

Proof:

If g is the cosine function, then

g(x) = cosx

g’(x) = lim△ x→0

g (x+△ x )−g(x)

△ x

= lim△ x→0

cos ( x+△ x )−cosx

△ x

= lim△ x→0

cos x cos (△ x )−sin x sin (△ x )−cos x

△ x

= lim△ x→0

cos x [cos (△ x )−1]

△ x - lim△ x→ 0

sin x sin (△ x )

△ x

= − lim

△ x→01−cos (△ x )

△ x( lim△ x→o

cos x ) ― ( lim△ x→osin x )

lim△ x→o

sin (△ x)

△ x

g’(x) = -0 · cosx – sinx · 1

g’(x) = - sin x

Theorem: Derivative of Tangent Function:D x (tanx )=sec2 x

Proof:

D x( tanx))= D x( sinxcosx )

= cosx ·D x ( sinx )−sinx · D x (cosx )

cos2 x

= (cosx ) (cosx )− (sinx ) (−sinx )

cos2 x

= cos2 x+sin2 xcos2 x

= 1

cos2 x

D x( tanx) = sec2 x

EXAMPLES:

1. f(x) = sin x

1−2cos x

f’(x) = ¿¿

= ¿¿

= cos x−2(cos2¿x+sin2 x )¿¿¿ ¿

f’(x) = cos x−2¿¿¿