Higher order derivatives and higher order differentials
      Higher derivatives of composite functions
         Higher derivatives of composite functions examples
      Higher derivatives of implicit functions
         Higher derivatives of implicit functions examples
Higher derivatives of composite functions
If  y = f (u) and u = g (x) such that  f is differentiable at  u and g is differentiable at x then, we differentiate the composite function  y [u (x)]  by applying the chain rule
y' (x) = y' (u) u' (x).
To get the second derivative of the given composite function we differentiate the above expression using both the product rule and the chain rule
y'' (x) = y' (u) u'' (x) + u' (x) [y'' (u) u' (x)] =  y'' (u) [u' (x)] 2 y' (u) u'' (x).
Proceeding the same way we get the third derivative of the composite function 
y''' (x) = y''' (u) [u' (x)] 3 3y'' (u) u' (x) u'' (x) y' (u) u''' (x and so on.
Higher derivatives of composite functions examples
Example:   Find the second derivative of
Solution:    
Example:   Find the second derivative of   y = sin2x.
Solution:     y'  = 2sin x cos x,     y''  = 2[sin x( -sin x) cos x cos x] = 2(cos2x - sin2x) = 2cos 2x.
Higher derivatives of implicit functions
The first derivative of a function  y = f(x) written implicitly as  F(x, y) = 0 is
In words, differentiate the implicit equation with respect to both variables at the same time such that, when differentiating with respect to x, consider y constant, while when differentiating with respect to y, consider x constant and multiply by y'.
We use this method to differentiate the above equation to extract the second derivative
and after substituting the value for  y'
Differentiating the second derivative equation above, applying the same method, we can get the third derivative of an implicit function and so on.
Higher derivatives of implicit functions examples
Example:   Find the second derivative of  y2 = 2px  or written implicitly  y2  - 2px = 0 as F(x, y) = 0.
Solution:   The first derivative,    2yy'  = 2p    or    yy'  = p,          y'  = p/y.
                 The second derivative,             (y' )2 yy''  = 0   =>    y'' - (y' )2 / y 
                    and by substituting      y'  = p/y,                             y'' - p2 / y3.    
We can obtain the same result using the above formulas. Therefore, first calculate
and plug these values into the formulas for the first and the second derivative
Example:   Find the second derivative of  y - tan-1y = x.
Solution:        
The second derivative,
 
Again, we can obtain the same result using the above formulas.
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