C) Then f (x) = sin xX(ln sinx + x cotx ) and gf(x) = cosxh
Derivatives of Some Standard Functions
(sec2x ln cos x - tan. 2 x) \dY - = f (x) + gf(x) dx ~ , = x("'), x > o d) Let f(x) = ( x ~ )g(x) If y = xX,lny = xln
(x) = x ( ~[xX-' ) + xXln x (1 + ln x)] Answer = f ' (x) + gf(x) = (xX)"[ h X+ x(1 + h ) ] + x(=) [$-I + xx-' 1m (1 + lnx)] d In sin x e) -(sin x)InX= (sin x)InX(ln x cot x + dx x d --(xX)= xX(l+Inx) dx In sin x Answer = (sin x)'"" (I" X Cot x + -X + x X ( l+ l n x ) dx dx = - a sin 8, - = a cos 8 a) d8 de 3 g'
)
E 10)
b) c) d)
z--dy
2a 1 2at t dy 3bsin28cos8 = - - tang - -dx -3acos2 8 sin8 a d~ a sin8 - sin 8 - -dx a(1- cos 8) (1 - cos8) -
dy . dy d) - cosx sin y - - sin x cosy - 2y - sin-'x dx dx + 2x2 sec2x = o
Y \r-T
sinx cosy + 3
dy - = dx
YL
- 4x tanx - 2x2 sec2 x
(1-x
- (cos x sin y + 2y sin-' x))
+
4x tan x