Lesson 3 derivative of hyperbolic functions

DIFFERENTIATION OF HYPERBOLIC FUNCTIONS

TRANSCENDENTAL FUNCTIONS

Kinds of transcendental functions:1.logarithmic and exponential functions2.trigonometric and inverse trigonometric functions3.hyperbolic and inverse hyperbolic functions Note:Each pair of functions above is an inverse to each other.

HYPERBOLIC IDENTITIES

DIFFERENTIATION FORMULA

Derivative of Hyperbolic Function

A. Find the derivative of each of the following functions and simplify the result:

x2 cosh x sinhy .1

)xcoshxsinh(xcosh'y

)xsinhx(coshxcosh)xcoshxsinh(xsinh'y

x cosh x coshx sinh x sinh'y

22

22

5

22

222

x hsecxy .2 x hsec)x tanhx hsec(x'y

xhsecy .3 2

x tanhx hsecx hsec2'y

x hsecx cothy .5

)xhcsc(x hsec)x tanhx hsec(xcoth'y 2

2x sinhlny .4

2

2

xsinh

xcosh x2'y

EXAMPLE:

)x tanh x1(x hsec'y

x tanhxhsec2'y 2

2xcoth x2'y

xhcscx tanhx cothx hsec'y 2

xhcsc1x hsec'y 2

hxcscxcothy

xcothx hsec'y'

2

xcoth lny .6 2

xcoth

x hcsc x coth2'y

2

2

xcotharccosy .7

xcoth1

xhcsc'y

2

2

xhcscxhcsc

xhcscxhcsc'y

22

22

x sinh

x coshxsinh

12

'y2

2

2

x sinhxcosh

2'y

x2 sinh

4'y

x2 hcsc4'y

xhcsc

xhcsc'y

2

2

xhcsc'y 2

)x h arctan(siny .8 2

22

2

x cosh

x cosh x2'y

2x hsec x2'y

22

2

x sinh1

x cosh x2'y

A. Find the derivative and simplify the result.

2xsinhxf .1

w4hsecwF .2 2

3 xtanhxG .3

3tcoshtg .4

x

1cothxh .5

xtanhlnxg .6

EXERCISES:

ylncothyf .7

xcoshexh .8 x

x2sinhtanxf .9 1

xxsinhxg .10

21 xtanhsinxg .11

0x,xxf .12 xsinh

Hyperbolic Functions Trigonometric Functions

1xsinhxcosh 22

xhsecxtanh1 22

xhcsc1xcoth 22

ysinhxcoshycoshxsinh)yxsinh(

ysinhxsinhycoshxcoshyxcosh

ytanhxtanh1

ytanhxtanhyxtanh

ytanxtan1

ytanxtanyxtan

ysinxsinycosxcosyxcos

ysinxcosycosxsinyxsin

xx 22 sectan1

1sincos 22 xx

xcsc1xcot 22

Identities: Hyperbolic Functions vs. Trigonometric Functions

Hyperbolic Functions Trigonometric Functions

Identities: Hyperbolic Functions vs. Trigonometric Functions

sinh 2x = 2 sinh x cosh x

2/1x2coshxsinh2

2/1x2coshxcosh2 xexsinhxcosh

xexsinhxcosh

2/x2cos1xcos2

2/x2cos1xsin 2

cos 2x = cos2x – sin2x

sin 2x = 2sinx cosx

cosh 2x = cosh2x +sinh2x