Derivatives of Products, Quotients, Sine, and Cosine

Derivative Formulas
There are two kinds of derivative formulas:
d
1.
Specific Examples: -x" or -
dx
2.
General Examples: (u +v) =u1+v1 and (cu) = cul (where c is a constant)
A notational convention we will use today is:
Proof of (u+v) = u +v . (General)
$@rt by using the definition af &hederivative.
(U +v) (x) = lirn (U+U)(X+ax) -(U+v)(x) Ax-0 ax U(X+ax) +V(X+AX)-U(X)-V(X)
= lirn
Ax-0 ax = lim {u(x +Ax) -u(x) + v (x +Ax) -v(x) AX-O Ax Ax
Follow the same procedure to prove that (cu) = cu .
Derivatives of sin x and cos x. (Specific) Last time, we computed
sinx
lim -= 1
x-0 x d sin(0 +Ax) -sin(0) sin( Ax)
-(sinx) IZEo = lirn = lim -= 1
dx AX-o ax AX-o ax
d COS(O +ax) -COS(O) COS(~X)i
-
-(co~x)I~=~= lim = lim = 0
dx AX-0 ax AZ-O ax
d d
So, we know the value of -sin x and of -cos x at x = 0. Let us find these for arbitrary x.
dx dx
d sin(x +Ax) -sin(x)
-sin x = lirn
dx AX-0 ax
Recall: sin x cos Ax + cos x sin Ax -sin(%) -lirn
Ax-0 Ax sin x(cos Ax -1) cos x sin Ax
= lim [ 1
AX-o Ax Ax
+
cos Ax -1 sin Ax
= AX-o Ax lim COSX(~)
lim sins( )+ asto
cos Ax -1 sin Ax
Since +0 and that --+ 1,the equation above simplifies to
Ax Ax
d
-sinx = cosx
dx A similar calculation gives
d
-cosx = -
dx
Product formula (General)
(uv) = ulv + uvl Proof: (uv)(X + AX)-(UV)(x) u(x + Ax)v(x + Ax) -u(x)v(x)
(uv) = lim = lim
hxt~ Ax Axto AX Now obviously,
so adding