1
c Roberto Barrera, Fall 2015
Math 142 4.1 Derivatives of Powers, Exponents, and Sums
Derivatives of Constants
For any constant c,
d
d
c = (c) =
dx
dx
Example: Find
1.
d
dx 2015
2.
d 1
dx 4
3.
d 2
dx π
Derivatives of Powers
If n is any real number (n may or may not be an integer) then
d n
d
x = (xn ) =
dx
dx
Example: Find
1.
d 2015
dx x
2.
d
dx
3.
d π
dx x
√
3 2
x
2
c Roberto Barrera, Fall 2015
Math 142 Derivatives of the exponential function
The exponential functin ex is its own derivative, that is,
d x
d
e = (ex ) =
dx
dx
Remark: This only works for ex ! The derivative of ax is NOT ax if a 6= e!
Derivatives of the natural logarithm
The derivative of ln(x) is
d
d
ln(x) = [ln(x)] =
dx
dx
Remark: This only works for base e! The derivative of loga (x) is NOT
Derivatives of constant times a function
Let c be any constant. If f is a function and f 0 exists, then
d
d
c f (x) = [c f (x)] =
dx
dx
Example: Find
1.
d
x
dx 4e
2.
d
dx π ln(x)
3.
d
−3
dx 3x
1
x
if a 6= e!
3
c Roberto Barrera, Fall 2015
Math 142 Derivatives of sums and differences
If f (x) and g(x) are functions and f 0 (x) and g0 (x) exist, then
d
dx [ f (x) ± g(x)]
exists and
d
[ f (x) ± g(x)] =
dx
Example:Find
1.
1
d 4
2)
(x
+
x
dx
2.
d
dx
ln(x) + 3x2
Rates of Change in Business
Need to know the effects of changes in production and sales on costs, revenues and
profits.
The word marginal refers to an instantaneous rate of change, that is, to the derivative.
C(x) cost function ⇒
R(x) revenue function ⇒
P(x) profit function ⇒
c Roberto Barrera, Fall 2015
Math 142 4
Example: A company has found that the cost (in dollars) from making x ovens is given
by
C(x) = .2x2 + 350.
Use the marginal cost to estimate the cost from making the 36th oven.