Differentiation of Composite Functions

Differentiation of composite functions is achieved using the Chain Rule, which allows you to differentiate a function that is composed of two or more functions. If you have a function within another function, the Chain Rule helps find the derivative of the overall composite function.

Chain Rule Explanation

• The Chain Rule states that if a function ( y = f(g(x)) ), where ( f ) and ( g ) are functions of ( x ), then the derivative of ( y ) with respect to ( x ) is given by:

  [ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) ]

  Here:

  • ( f'(g(x)) ) is the derivative of the outer function evaluated at the inner function.
  • ( g'(x) ) is the derivative of the inner function.

Steps for Differentiating Composite Functions

1. Identify the Inner and Outer Functions:
   • The inner function is the one inside another function, i.e., ( g(x) ).
   • The outer function is the one that wraps around the inner function, i.e., ( f ).
2. Differentiate the Outer Function:
   • Take the derivative of the outer function, leaving the inner function unchanged.
   • This gives you ( f'(g(x)) ).
3. Differentiate the Inner Function:
   • Now, differentiate the inner function ( g(x) ) to get ( g'(x) ).
4. Multiply the Two Derivatives:
   • Multiply the derivative of the outer function by the derivative of the inner function to get the final result: ( f'(g(x)) \cdot g'(x) ).

Example 1: Differentiating ( y = (3x + 2)^4 )

• Step 1: Identify the inner function and the outer function.
  • Inner function, ( g(x) = 3x + 2 )
  • Outer function, ( f(u) = u^4 ), where ( u = g(x) )
• Step 2: Differentiate the outer function with respect to ( u ):
  • ( f'(u) = 4u^3 )
• Step 3: Differentiate the inner function with respect to ( x ):
  • ( g'(x) = 3 )
• Step 4: Apply the Chain Rule: [ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) = 4(3x + 2)^3 \cdot 3 = 12(3x + 2)^3 ]

Example 2: Differentiating ( y = \sin(5x^2) )

• Step 1: Identify the inner and outer functions.
  • Inner function, ( g(x) = 5x^2 )
  • Outer function, ( f(u) = \sin(u) ), where ( u = g(x) )
• Step 2: Differentiate the outer function:
  • ( f'(u) = \cos(u) )
• Step 3: Differentiate the inner function:
  • ( g'(x) = 10x )
• Step 4: Apply the Chain Rule: [ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) = \cos(5x^2) \cdot 10x = 10x \cos(5x^2) ]

Example 3: Differentiating ( y = e^{\cos(x)} )

• Step 1: Identify the inner and outer functions.
  • Inner function, ( g(x) = \cos(x) )
  • Outer function, ( f(u) = e^u ), where ( u = g(x) )
• Step 2: Differentiate the outer function:
  • ( f'(u) = e^u )
• Step 3: Differentiate the inner function:
  • ( g'(x) = -\sin(x) )
• Step 4: Apply the Chain Rule: [ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) = e^{\cos(x)} \cdot (-\sin(x)) = -\sin(x) e^{\cos(x)} ]

The Chain Rule simplifies the process of differentiating composite functions by breaking them down into manageable parts. This method is essential when dealing with nested functions in calculus.