Derivatives

Derivatives (for each below, left side = right side)

Question	Answer
f'=df/dx	lim h->0 [f(x+h)-f(x)]/[(x+h)-x]
f'=df/dx	lim h->0 [f(x+h)-f(x-h)]/[(x+h)-(x-h)]
(c f)'	c f' (when c is a constant)
(f + g)'	f' + g'
(f g)'	f' g + f g'
	memorize

 

Question	Answer
(f g h)'	f' g h + f g' h + f g h'
(1/f)'	-f'/f2 (for f≠0)
(f/g)'	(f' g - f g')/g2 (for g≠0)
(fg)'	fg (g' ln f + g f'/f) (for f≠0)
c'	0
x'	1
	memorize

 

Question	Answer
f(g(x))'	(df/dg)(dg/dx)
g(f(x))'	(dg/df)(df/dx)
f(g(h(x)))'	(df/dg)(dg/dh)(dh/dx)
f(h(g(x)))'	(df/dh)(dh/dg)(dg/dx)
g(f(h(x)))'	(dg/df)(df/dh)(dh/dx)
g(h(f(x)))'	(dg/dh)(dh/df)(df/dx)
h(f(g(x)))'	(dh/df)(df/dg)(dg/dx)
h(g(f(x)))'	(dh/dg)(dg/df)(df/dx)
	memorize

 

Question	Answer
(c x)'	c
|x|'	x/|x|=sgn x for x≠0
(xc)'	c xc-1
(1/x)'=(x-1)'	-x-2=-1/x2
(1/xc)'=(x-c)'	-c x-c-1=-c/xc+1
(√x)'=(x1/2)'	(1/2) x-1/2=1/(2 √x) for x>0
	memorize

 

Question	Answer
(cx)'	cx ln c for c>0
(ex)'	ex
(logcx)'	1/(x ln c) for c>0, c≠1
(ln x)'	1/x for x>0
(ln |x|)'	1/x
(xx)'	xx (1 + ln x)
(Γ(x))'	∫0∞ tx-1 e-t ln t dt
	memorize

 

Question	Answer
(sin x)'	cos x
(cos x)'	-sin x
(tan x)'	sec2x = 1/cos2x
(sec x)'	sec x tan x
(csc x)'	-csc x cot x
(cot x)'	-csc2x = -1/sin2x
	memorize

 

Question	Answer
(arcsin x)'	1/√1 - x2
(arccos x)'	-1/√1 - x2
(arctan x)'	1/(1 + x2)
(arcsec x)'	1/[|x| √x2 - 1]
(arccsc x)'	-1/[|x| √x2 - 1]
(arccot x)'	-1/(1 + x2)
	memorize

 

Question	Answer
(sinh x)'	cosh x = (ex + e-x)/2
(cosh x)'	sinh x = (ex - e-x)/2
(tanh x)'	sech2x
(sech x)'	-tanh x sech x
(csch x)'	-coth x csch x
(coth x)'	-csch2x
	memorize

 

Question	Answer
(arcsinh x)'	1/√x2 + 1
(arccosh x)'	1/√x2 - 1
(arctanh x)'	1/(1 - x2)
(arcsech x)'	-1/[x √1 - x2]
(arccsch x)'	-1/[x √1 + x2]
(arccoth x)'	1/(1 - x2)
	memorize