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¶

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., \( f \).
Differentiate the Outer Function:
Take the derivative of the outer function, leaving the inner function unchanged.
This gives you \( f'(g(x)) \).
Differentiate the Inner Function:
Now, differentiate the inner function \( g(x) \) to get \( g'(x) \).
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.