Hyperbolic,

Inverse hyperbolic

Derivative of hyperbolic & Inverse hyperbolic

Successive differentiation

Leibnitz’s Theorem

Definitions of Hyperbolic

functions sinh

2

x xe ex

cosh

2

x xe ex

2 2

2 2cosh sinh 12 2

x x x xe e e ex x

sinhtanh

cosh

x x

x x

x e ex

x e e

coshth

sinh

x x

x x

x e eco x

x e e

1 2sech

cosh x xx

x e e

1 2csch

sinh x xx

x e e

Derivatives of Hyperbolic

functions sinh cosh

2

x xd e ex x

dx

cosh sinh

2

x xd e ex x

dx

2tanh secd

x h xdx

2th cscd

co x h xdx

sech sec tanhd

x hx xdx

csch csc cothd

x hx xdx

Derivatives of inverse

hyperbolic functions 1 1

2 2

1 1sinh cosh

1 1

d du d duu and u

dx dx dx dxu u

1 1

2 2

1 1tanh coth

1 1

d du d duu and u

dx dx dx dxu u

1 1

2 2

1 1sec csc

1 1

d du d duh u and h u

dx dx dx dxu u u u

Indeterminate

forms.

0

0

0

0

0

1

0

Determinate forms

0

0

0

0

0

1

0

0

0

1

-

a) cosh (-x) = cosh x i.e. cosh x is an even function of x

sinh(-x) = sinh x i.e. sinh x is an odd function of x cosech (-x) = - cosech x tanh(-x) = - tanh x coth(-x) =-coth x sech(-x) = sech x b) cosh x + sinh x =

Properties of hyperbolic functions.

Successive Dlfferentiation

Let y=f(x) be a function of x. Its derivative w.r.t. ‘x’ is denoted by dy/dx or y1 or f’(x). This is called first

derivative of y w.r.t.x.

The derivative of first derivative of y is called second derivative of y and is denoted by

d 2y/dx2 or y2 or f’(x)

The derivative of second derivative of y is called third derivative of y and is denoted by d3y/dx3 or

y3 or f’’’ (x) so on

The derivatives f’(x) , f’’(x), f’’’(x) ……are called successive derivatives of f.

dny/dxn or yn or fn(x) is called derivative of nth order.

Leibnitz’s Theorem

If y=uv, where u and v are two function of x

having derivatives of nth order, then

Where suffixes of u and v denote the

differentiations w.r.t.x and denote the

number of combinations if n differents

yhings taken r at a time.

• INDUCTION METHOD

• STEP 1: to prove the result for n=1 i.e.

• i.e.

• By actual differentiation of y=uv

• We have

which proves that the theorem is true for

n=1

Step ii

Suppose the result is true for n=m

i.e

Step III To prove the result for n=m+1

Differentientiate both sides of step II w.r.t.x

• =

• Now

• Putting r=1,2,3,…. In last result we get

• Which proves that this theorem is true for

n=m+1

• Hence , by induction the theorem is true

vnN

• Find the nth derivation of

• (i) cos x

• (ii)