Lesson Taking derivative of a function, which is defined implicitly

Taking derivative of a function, which is defined implicitly

Problem 1

Let   y = sin (t),   x = ln (t).   It defines  y  as a function of  x,   y = y(x),   implicitly. 


       d^2 y
Find  ------- when  x = .
        dx^2

If  x = ln(t),  then  

    t = ,      (1)

where "e" is the base of natural logarithms.


Therefore, in this problem, after making substitution (1), we have

    y = sin(e^x),     (2)

i.e. function y is expressed as the composition of function sine and exponent.


So, we apply the formula for the derivative of a composite function and find 
first derivative of y with respect to x

     =  = .    (3)


Then we find second derivative as the derivative of (3)

    d^2 y
   ------- (x) =  +  =  + .
    dx^2


Now we substitute x =  to get

    d^2 y
   -------  =  + .
    dx^2


To get the value, use in calculations approximate values e = 2.71828, pi = 3.14159.


    d^2 y
   -------  =  +  = 479.18428  (rounded).
    dx^2