C12-16 Flashcards
differentiating a constant
equals 0
differentiating a linear
equals a constant
differentiating a square
equals a linear
differentiating sin(x)
equals cos(x)
if f and g are differentiable at a then ?
then f+g is differentiable at a and
f+g)’(a) = f’(a) + g’(a
what is the product rule?
f’ g + f g’
if f(x) = x^n for nEN then
f’(x) = na^(n-1)
what is the quotient rule?
(f’ g - f g’)/ g^2
f= sin(x)
f’ = cos(x)
f
cos(x)
f’
-sin(x)
f
tan(x)
f’
sec^2(x)
f
sec(x)
f’
sec(x) tan(x)
f
cosec(x)
f’
-csc(x) cot(x)
f
cot(x)
f’
-csc^2(x)
considering function f on an interval I on which it is differentiable and invertible on I by f^(-1)
for all bE I where f’(b) does not equal 0, the inverse function f^(-1) is then differentiable at a= f(b) and
(f^-1)’ (a) = 1/(f’(b)) = 1/ (f’(f^-1(a))
f
sin(x)
f^(-1)
arcsin(x)
arcsin’ (x)
1/ (1-x^2) ^1/2
arccos’(x)
-1/ (1-x^2) ^1/2
arctan’(x)
1/ 1+ x^2
what is the definite integral?
number I which is denoted by
I= (integral from a to b) f
log (base a) b =x means what
b= a^x
log(base a) 1 =?
0
what is the natural logarithm defined by
ln(x) = (integral from 1 to x) 1/t dt
what is the exponential function
exp(x) =y implies x= ln y
exp’(x) =?
exp(x)
e^x =?
exp(x)
a^x = ? (exponential)
e^(x lna)
d/dx ln (f(x)) =?
f’(x) / f(x)
what are the hyperbolic functions?
sinh (x) cosh(x) ect …
cosh^2(x) - sinh^2(x) =?
1
what is the equation for integration by parts?
(int) fg’ = fg - (int) f’g
beneficial when (int) f’g is easier than initial
