Page 1
Math 151-copyright Joe Kahlig, 11C
Section 4.1: Exponential Functions and their Derivatives
Definition: An exponential function is of the form f (x) = ax where a is a positive constant and is
defined in the following manner.
If x = n (positive integer then an = a ∗ a ∗ a ∗ ... ∗ a and a−n =
1
an
If x = 0 then a0 = 1
√
p
If x is a rational number, x = , with p and q integers and q > 0, then ap/q = q ap
q
If x is an irrational number then we define ax = lim ar where r is a rational number.
r→x
Three cases for y = ax
Exponential Rules
ax+y = ax ay
ax−y =
Example: Evaluate these limits.
A) lim 3 − 2
x→∞
x
B) lim 3 − 2
x→−∞
π
4
=
x
π
4
=
ax
ay
(ax )y = axy
(ab)x = ax bx
Math 151-copyright Joe Kahlig, 11C
C) lim
x→∞
D) lim
80
4 + 2e−0.15x
x→−∞
80
4 + 2e−0.15x
2e3x + e−2x
x→∞ 3e4x + 5e−2x
E) lim
2e3x + e−2x
x→−∞ 3e4x + 5e−2x
F) lim
Page 2
Math 151-copyright Joe Kahlig, 11C
2e5x + 6e−3x
x→0 4e−5x + e2x
G) lim
Derivatives of Exponentials
Find the derivative of f (x) = ax
Example: Find the indicated derivative.
A) y ′ if y = e5x + 5xe
B) y ′ if y = cot(e3x )
Page 3
Math 151-copyright Joe Kahlig, 11C
C) y ′ if y = ecos(6x)
D) y ′′′ if y = ex
2
Example: Find the equation of the tangent line at the point (0, 3) for
3exy = x2 + y
Page 4
Math 151-copyright Joe Kahlig, 11C
Example: Find the value(s) of A that make y = eAx a solution to
8y + 10y ′ = 3y ′′
Page 5