Differential calculus - derivatives
      The chain rule
         Differentiation using the chain rule, examples
The chain rule
If  y = f (u)  and  u = g (x)  such that f is differentiable at  u = g (x)  and g is differentiable at x, that is,
then, the composition of f with g,
 y = ( f o g )(x) =  f [g (x)]
is differentiable at x, and then
   
Thus, the derivative of a composition of functions is equal to the derivative of outside function, with respect to
the inside function (i.e., taking it as the independent variable), times the derivative of the inside function.
Differentiation using the chain rule, examples
Example: Find the derivative of the function  y = (x2 - 3x + 5)3.
Solution:   Consider  y = f [g (x)]    where,    y = f (u) = u3    and    u = g (x) = x2 - 3x + 5
so that,    
Therefore,    
Example: Find the derivative of the function  y = sin3 2x = (sin 2x)3.
Solution:   Consider  y = f {g [h(x)]}  where,   y = f (u) = u3,   u = g(v) = sin v  and   v = h(x) = 2x.
so that,  
Therefore,  
Example: Find the derivative of the function 
Solution:   Consider  y = f {g [h (x)]}  where,
Therefore,  
Example: Find the derivative of the function 
Solution:  
  
Remember, if  
Example: Find the derivative of the function 
Solution:  
  
Example: Find the derivative of the function 
Solution:  
  
Remember, if
therefore, if
 
Calculus contents C
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