4.1 - 4.2: Basic rules for derivatives
June 9th 2015
The Basic Derivative Rules (using Leibniz notation):
a) Derivative of a constant:
d
c=
dx
b) Derivative of a power xn (n is any real number, may or may not be an integer):
d n
x =
dx
c) Derivative of ex :
d x
e =
dx
d) Derivative of a constant times a function:
d
(cf (x)) =
dx
e) Derivative of lnx:
d
lnx =
dx
f) Derivative of sum or difference:
d
(f (x) ± g(x)) =
dx
g) Derivative of a product of two functions
d
(f (x)g(x)) =
dx
h) Derivative of a quotion of two functions
d
dx
f (x)
g(x)
Example 1. Find the derivatives of:
a) e.
b) π.
√
c) x
2
−
1
x
√
2
.
3
d) x 4
1
=
e) 2ex
f ) elnx
g) ex lnx
h)
√
x − x3
i) x3 ex
j)
ex
x2 +1
k)
2u−1
u3 −1
l)
3−2ex
1−2x
m)
lnx
−x
x2
2
Example 2. Find the equation of the tangent line to the graph of
f (x) =
x+4
x
at the point where x = 1.
Example 3. Find the point on the graph of
f (x) = x2 − 3x + 2
such that the tangent line of the graph at that point is:
1. horizontal
2. parallel to the line y = 2x − 1
3. perpendicular to the line y = 2x − 1
Example 4. The position of a ball is governed by the equation s(t) = 2t3 + t2 − 7t + 1, where t in
seconds and s in meters.
a) Calculate the velocity of the ball at time t
3
b) Compute the ball’s velocity after 4 s
c) When is the ball at rest (v(t) = 0)?
Marginal Analysis
Definition: If C(x), R(x), and P (x) are the cost, revenue, and profit functions, respectively, then the
instantaneous rate of change i.e. derivatives C 0 (x), R0 (x), and P 0 (x) are the marginal cost, marginal
revenue, and marginal profit.
4
Example 5. In order to produce x units, a company cost C(x) = 20, 000 + 7x + 0.5x2 .
a) Calculate the marginal cost function
b) When 2000 units are produced, find the marginal cost
c) Compute the cost of producing 2001 units
5