Derivative of f(g(x))

We now explore the derivative of the composite function f(g(x)). It is commonly represented as d/dx [f(g(x))] or [f(g(x))]', and its value depends on the derivatives of both f and g. The function f(g(x)) is differentiable within its domain if both f(x) and g(x) are differentiable. The key concepts are mentioned below: Composite Function: A function composed of two functions, f and g, where f(g(x)) is the composition. Chain Rule: A rule for differentiating composite functions. Derivative: Represents the rate of change of a function concerning its variable.

The derivative of f(g(x)) is denoted as d/dx [f(g(x))] or [f(g(x))]'. The formula used to differentiate f(g(x)) is: d/dx [f(g(x))] = f'(g(x)) · g'(x) This formula applies to all x where both f and g are differentiable.

We can derive the derivative of f(g(x)) using various proofs. To show this, we will use differentiation rules. Some methods for proving this include: By Chain Rule By Substitution We will now demonstrate that the differentiation of f(g(x)) results in f'(g(x)) · g'(x) using these methods: Using Chain Rule To prove the differentiation of f(g(x)) using the chain rule: Consider f(x) and g(x), where the composite function is f(g(x)). By the chain rule: d/dx [f(g(x))] = f'(g(x)) · g'(x) This formula states that to differentiate a composite function, multiply the derivative of the outer function evaluated at the inner function by the derivative of the inner function. By Substitution Another method involves substitution: Let h(x) = g(x), so f(g(x)) = f(h(x)). The derivative is: d/dx [f(h(x))] = f'(h(x)) · h'(x) Substituting h(x) back for g(x), d/dx [f(g(x))] = f'(g(x)) · g'(x) Hence, proved.

When a function is differentiated multiple times, the derivatives obtained are higher-order derivatives. To understand them, consider a scenario where the speed (first derivative) and the rate of change of speed (second derivative) change. Higher-order derivatives provide insights into the behavior of functions like f(g(x)). For the first derivative, we write f′(x), indicating the function's rate of change or slope. The second derivative, f′′(x), is derived from the first derivative, indicating the rate of change of the rate of change. This pattern continues for higher-order derivatives. For the nth derivative of f(g(x)), we generally use fⁿ(x) to represent the nth derivative, which tells us about the change in the rate of change.

When g(x) has points where its derivative is undefined, the derivative of f(g(x)) is also undefined at those points. If g(x) is a constant, the derivative of f(g(x)) depends solely on the derivative of the constant function.

Chain Rule: A rule for differentiating composite functions, expressed as d/dx [f(g(x))] = f'(g(x)) · g'(x). Composite Function: A function formed by applying one function to the results of another, denoted as f(g(x)). Higher-Order Derivative: Derivatives obtained by differentiating a function multiple times, such as the second, third, or nth derivative. Differentiable: A function is differentiable at a point if it has a defined derivative at that point. Rate of Change: The rate at which a function changes concerning its variable, often determined by its derivative.