Derivatives Flashcards
Limit definition of a derivative
f’(x) = lim(h->0) (f(x + h) - f(x))/h if the limit exists
m = f’(a) is the slope of?
The tangent line to y = f(x) at x = a
Equation of the tangent line
y = f(a) + f’(a)(x-a)
The tangent line can be?
Instantaneous rate of change at a point, velocity
a’
0
[f(x) + g(x)]’
f’(x) + g’(x)
[f(x)g(x)]’
f(x)g’(x) + f’(x)g(x)
[f(g(x)]’
f’(g(x))g’(x)
[cf(x)]’
cf’(x)
[f(x) - g(x)]’
f’(x) - g’(x)
[f(x)/g(x)]’
(f’(x)g(x) - f(x)g’(x))/(g(x))^2
[f^-1(x)]’
1/(f’(f^-1(x)))
[x^a]’
ax^(a-1)
[e^x]’
e^x
[e^u]’
u’e^u
[a^x]’
a^(x)lna
[a^u]’
a^(u)u’lna
[lnx]’
1/x
[lnu]’
u’/u
[log(a)x]’
1/xlna
[log(a)u]’
u’/ulna
[sinx]’
cosx
[cosx]’
-sinx
[tanx]’
sec^(2)x
[cotx]’
-csc^(2)x
[secx]’
secxtanx
[cscx]’
-cscxcotx
[sinu]’
u’cosu
[cosu]’
-u’sinu
[tanu]’
u’sec^(2)u
[cotu]’
-u’csc^(2)u
[secu]’
u’secutanu
[cscu]’
-u’cscucotu
[arcsinx]’
1/sqrt(1-x^2)
[arccosx]’
-1/sqrt(1-x^2)
[arctanx]’
1/(1+x^2)
[arccotx]’
-1/(1+x^2)
[arcsecx]’
1/|x|sqrt(x^(2) - 1)
[arccscx]’
-1/|x|sqrt(x^(2) - 1)
[arcsinu]’
u’/sqrt(1-u^2)
[arccosu]’
-u’/sqrt(1-u^2)
[arctanu]’
u’/(1 + u^2)
[arccotu]’
-u’/(1 + u^2)
[arcsecu]’
u’/|u|sqrt(u^(2) - 1)
[arccscu]’
-u’/|u|sqrt(u^(2) - 1)
