Derivatives of Exponential, Logarithmic, and Hyperbolic Functions

I.  Quiz

II.  Homework

III.  The Derivative of f(x) = e^x

We will investigate the derivative of y = e^x by sketching the graph.  Then the students will complete the table:

x	ex	(ex )'
-1	0.368
0	1
1	2.718
2	7.389
3	20.08

Definition:  e is the unique number such that if f(x) = e^x  then f'(x) = e^x .

Exercises: Find f'(x) if f(x) =

A)  3e^x

B)  xe^x

C)  e^x /x

D)  e^7x

E)  xe^3ex

Note that if f(x) = e^x then f'(x) = lim h -> 0 [e^x+h - e^x ]/h.   In particular if x = 0 then

lim h-> 0 (e^h -1)/h = 1

IV.  Derivatives of Other Exponential Functions

To find the derivative of f(x) = 2^x we use the formula

2^x = e^xln2  

and the chain rule gives us

f'(x) = e^x ln 2

In particular if f(x) = b^x

then f'(x) = e^x ln(b)

V)  The Derivative of the Natural Logarithm Function

Recall that the natural logarithm function g(x) = ln(x) is the inverse of the exponential function f(x) = e^x .  To find f'(x) we use the inverse formula:

g'(x) = 1/f'(g(x)) = 1/e^lnx  = 1/x.

Theorem:  If f(x) = ln x then f'(x) = 1/x

Example:  Find f'(x) if

f(x) = e^x ln(x)

Solution:  We use the product rule

f'(x) = e^x ln x + e^x /x

VI.  Hyperbolic Functions

Exercises:  Find f'(x) if

A)  cosh(x) = [e^x + e^-x]/2

B)  sinh(x) = [e^x - e^-x]/2

C)  tanh(x) = sinh(x)/cosh(x)

D)  coth(x) = cosh(x)/sinh(x)

E)  sech(x) = 1/cosh(x)

F)  csch(x) = 1/sinh(x)

Proof that cosh²(x) - sinh²(x) = 1:

Take derivatives of the left side:

2cosh(x)sinh(x) -2sinh(x)cosh(x) = 0

Hence the left hand side is a constant.  Plugging in x = 1 gives

 cosh²(1) - sinh²(1) = 1.