f (x) = y = arcsin x find y’ f (x) = y = arctan x find y’

f (x) = y = arcsec x find y’ Formulas:

2 2 2

2 2 2

2 2 2

d u du uu C

dx a1 u a u

d u du 1 uu C

dx a a1 u a u

ud u du 1u C

dx a au u 1 u u a

'arcsin arcsin

'arctan arctan

'arc sec arc sec

= = +⎡ ⎤⎣ ⎦− −

= = +⎡ ⎤⎣ ⎦ + +

= = +⎡ ⎤⎣ ⎦− −

∫

∫

∫

Give the domain of and compute its derivative.

( ) ( )( )x xf arctan ln=

Give the domain of ( )xe

x2

g arcsin= and find the equation for the

tangent line to the graph of this function at x = 0.

#12. Differentiate: 1y xtan−= #16. Differentiate: ( ) x1x e tanf

−=

#20. Differentiate: 1 2y x 2sec −= +

#40. Evaluate: 1

21

dx

1 x− +∫

#42. Evaluate: 1

0 2

dx

4 x−∫

#46. Evaluate: ( )

5

22

dx

9 x 2+ −∫

4

xdx

1 x−∫

Watch for details. These look similar.

2

1dx

16 x+∫

2

xdx

16 x+∫

Hyperbolic Functions

Section 7.8

Primary Definitions

Hyperbolic cosine: Hyperbolic sine:

( )

( )

x x

x x

e ex

2

e ex

2

cosh

sinh

−

−

+=

−=

( ) ( )( ) ( ) ( )

( )

( ) ( ) ( ) ( )

x xx x

x x

1 1x x

x x

sinh coshtanh coth

cosh sinh

sech cschcosh sinh

= =

= =

Why are these called Hyperbolic Functions? Recall that sine and cosine are called circular functions. Remember the unit circle?

−3 −2 −1 1 2 3

−5

5

10

−3 −2 −1 1 2 3

−5

5

10

( )

( )x

x

x

xex

2

e e

e

x2

s

s

inh

co h−

−

+

−

=

=

( )

( ) x

x

x

x e

eg

f

−=

=

Hyperbolic Identity Show that

( ) ( )2 2t t 1cosh sinh− =

sinh (–x) = – sinh x (odd) cosh (–x) = cosh x (even)

cosh x + sinh x = cosh x – sinh x =

( )dx

dxsinh =

( )dx

dxcosh =

Derivatives and Integral formulas:

2

2

dx x

dx

dx

dx

dx x

dx

dx x x

dx

dx x

dx

dx x x

dx

sinh cosh

cosh sinh

tanh sec h

sec h tanh sec h

coth csc h

csc h coth csc h

=

=

=

= −

= −

= −

1. ( )( )dx

dxtanh ln⎡ ⎤

⎢ ⎥⎣ ⎦

2. ( )( )xd

dx 1 x

cosh

sec h

⎡ ⎤⎢ ⎥

+⎢ ⎥⎣ ⎦

3. ( ) ( )( )x x dxtanh ln cosh =∫

4. ( )2 2x x dxsec h =∫

5. Find the area bounded by the graphs of and for .

( )x xf s inh=( )x xg cosh= 0 x 10ln≤ ≤

1 2 3

5

6.

( )2 3x dxcosh∫

7.7

Differentiate: