Marginal Cost Function - if C( x )
function
x equals units produced
Marginal Derivative
*
C ' ( x) is the marginal cost
C '( x) measures the actual cost incurred by producing an additional commodity
One-Sided Limits
Right hand limit - exists if the limit can be made as close to the number L as we please
by taking x sufficiently close to, but not equal to a, on the right side of a
Left hand limi
Limit - the function f has the limit L as x approaches a if the value of f (x) can be made as
close to the number L as we please by taking x sufficiently close to, but not equal to a
lim f ( x )=L
x a
Product Rule - If f and g are differentiable at x, then so is the product of
1
1
f g and
1
d /dx [ f ( x )g ( x)]=f ( x ) d /dx g( x )+ g (x) d /dx f (x ) or f g =f g + f g
Quotient Rule - If f and g
Average Rate of Change -
(f (b)f (a)/(ba)
Instantaneous Rate of Change -
lim (f (x+ x)f (x )/ x
x 0
Derivative - the slope of a line tangent to a point, x
Derivative Notation
D x f ( x)
dy /dx
y 1
f 1
Derivatives of Exponential Functions
d /dx e x =e x
f (x)
1
f (x)
d /dx (e )=f ( x) e
Derivatives of Logrithms
d /dx ln x=1/ x
d /dx [ln f ( x)]=f 1 ( x)/f ( x) ,
f ( x)>0
f ( x)=g [h (x)] , then f 1 ( x )=d /dx g[h( x )] . Equivalently, if we write
y=f (u)=g(u) , where u=h(x) , dy /dx=[g (u)/g ( x)][h(u)/h (x)]
Chain Rule - if
n
General Power Rule - if f is differentia
Rules of Differentiation
1) Derivative of a constant is 0
2) Power Rule - d /dx ( x n)
n x n1
3) Constant Multiple of a Function - d /dx [cf ( x )]=c d /dx f ( x )
4) Sum or Difference - d /dx [f ( x