Page 18 - mathematics

Table (4) Example

Inverse Trigonometric & Hyperbolic * Differentiate the functions:
Functions

cot1 x tan1 x tanh1 x co-th1 1 x  a) y  sin1 x 2 y '  1 2x 
1 tan
1 1 x 2 1  1 x 2 2
1 x 2 1 x 2 1x 2
cosh1 x  b ) y  csc1 sinx y '  1 cosx 
cos1 x sin 1 x sinh 1 x 1
1 sinx  sinx 2 1
1 1 1
 1 sin x 2 1 x c) y   sin1 ex 4
1 x 2 1 x 2 y'   ex 3
1 x 2 cs-c1h1 x 
sec1 x se-ch1 1 x
csc1 xsec x 1 x 2 4  sin1  1  1 e x
1 x 1 x 2  1ex
1 2 ex
x x 2 1
x x 2 1

Example Implicit Differentiation

* Differentiate the function: * If x can be expressed in the form y = f(x), then we say
that x is an explicit function of y. In some cases there
y sinh1  e sin 1 x are a relation of x and y which can not be expressed in
  the above form. These relations are called implicit
 functions.

 Steps of Implicit Differentiation:

1 * Differentiate both sides of the equation with respect
1  e sin1 x
   y '  to x. Remember that y is a function of x.
 e sin1 x
* Solve the differentiated equation for dy/dx in terms
2
of x and y.
 1 1

 1 x 2 2 x

18