Derivative of Composite Function

To understand the derivative of a composite function, we consider two functions, f(x) and g(x), where f(g(x)) is the composite function. The derivative is typically represented using the chain rule: d/dx [f(g(x))] = f'(g(x)) · g'(x). This rule allows us to differentiate complex functions by breaking them down into simpler parts. Key concepts include: - Composite Function: A function consisting of two functions, f and g, such that f(g(x)). - Chain Rule: A fundamental differentiation rule for composite functions. - Differentiability: The property that indicates whether a function has a derivative at each point in its domain.

The formula for the derivative of a composite function is given by the chain rule: d/dx [f(g(x))] = f'(g(x)) · g'(x). This formula applies whenever both f and g are differentiable functions.

The derivative of a composite function can be proven using the chain rule. We demonstrate this using different methods: - By First Principle - Using the Chain Rule - Using the Product Rule We will now derive the differentiation of a composite function using these methods: By First Principle To prove using the first principle, consider f(g(x)). Its derivative by limit definition is: (f(g(x)))' = limₕ→₀ [f(g(x + h)) - f(g(x))]/h. Using the chain rule, we express this as: (f(g(x)))' = f'(g(x)) · g'(x). This result follows from recognizing the limits and applying the chain rule. Using Chain Rule The chain rule directly gives us the derivative: d/dx [f(g(x))] = f'(g(x)) · g'(x). This is the simplest and most direct method to differentiate composite functions. Using Product Rule Although not typically used for composite functions, the product rule can sometimes assist in understanding derivative relations within parts of the composite function. We consider: Let u = f(g(x)) and v = g(x), then: d/dx [u·v] = u'·v + u·v'. While this is less common for composite functions, it highlights the interrelation within the function parts.

When a composite function is differentiated several times, the resulting derivatives are called higher-order derivatives. These derivatives can be complex, as they require repeated application of the chain rule. For instance, the second derivative of a composite function involves: d²/dx² [f(g(x))] = [f''(g(x)) · (g'(x))²] + [f'(g(x)) · g''(x)]. Higher-order derivatives provide deeper insights into the behavior of composite functions.

Certain points or conditions might make the derivative of a composite function undefined or unique: - If g(x) is constant, the derivative f(g(x)) becomes zero. - If f is not differentiable at g(x), the derivative does not exist.

Differentiating composite functions can be tricky. Here are common mistakes and how to avoid them:

Composite Function: A composition of two functions, represented as f(g(x)). Chain Rule: A rule used to differentiate composite functions. Differentiability: A function's ability to have a derivative at each point in its domain. Higher-Order Derivatives: Derivatives taken beyond the first derivative, often requiring repeated application of differentiation rules. Inner Function: The function g(x) within the composite function f(g(x)), which is differentiated first using the chain rule. ```