derivatives Flashcards
derivative of position
velocity
derivative of velocity
acceleration
what is instantaneous velocity
s’(t)
what is average velocity
distance/ time
how do you know when the particle is speeding up or slowing down
it is speeding up when v(t) and a(t) have the same sign. it is slowing down when they have opposite signs
when is the particle moving from left to right
when v(t)>0
when is the particle moving from right to left
when v(t)<0
what is the derivative of sinx
cosx
what is the derivative of cosx
-sinx
what is the derivative of tanx
sec^2x
what is the derivative of cotx
-csc^2x
what is the derivative of secx
secxtanx
what is the derivative of cscx
-cscxcotx
chain rule
derivative of outer x derivative of inner
product rule
(first)(derivative of second)+(second)(derivative of first)
quotient rule
((bottom)(derivative of top)-(top)(derivative of bottom))/(bottom)^2
inverse function theorem
(f^-1)’ = 1/ f’(f^-1(x))
derivative of inverse sin x
1/ √(1-x^2)
derivative of inverse cos x
-1/ √(1-x^2)
derivative of inverse tan x
1/ (1+x^2)
derivative of inverse cot x
-1/ (1+x^2)
derivative of inverse sec x
1/ lxl√(x^2-1)
derivative of inverse csc x
-1/ lxl√(x^2-1)
implicit differentiation steps
- derivative of both sides
- move all of the y’ to one side
- solve for y’
- if applicable, plug in the given
lnx=
logeX
logx
log10X
derivative of e^x
e^x
derivative of lnx
1/x
derivative of b^x
B^xlnb
derivative of logbX
1/(xlnb)
derivative of e^u
e^u*u’
derivative of b^u
b^u*u’lnb
derivative of lnu
u’/u
