Derivative of a Composite Function
The derivative of a composite function is a fundamental concept in calculus that allows us to find the rate of change of a function composed of two or more functions. In this article, we will discuss the definition of a composite function, the chain rule, and how to apply it to find the derivative of a composite function.
Definition of a Composite Function
A composite function is a function that is composed of two or more functions. For example, if we have two functions f(x) and g(x), the composite function, denoted as (f o g)(x), is defined as:
(f∘g)(x)=f(g(x))
In other words, we apply the function g to the input x, and then apply the function f to the output of g(x).
The Chain Rule
The chain rule is a rule in calculus that allows us to find the derivative of a composite function. It states that if we have a composite function (f o g)(x), the derivative of the composite function can be found by multiplying the derivative of the outer function f'(g(x)) with the derivative of the inner function g'(x).
Mathematically, the chain rule can be written as:
(f∘g)′(x)=f′(g(x))⋅g′(x)
In other words, we first find the derivative of the outer function f'(g(x)), evaluated at g(x), and then multiply it with the derivative of the inner function g'(x).
Example
Let's consider an example to illustrate the application of the chain rule. Suppose we have a composite function h(x) defined as:
h(x)=sin(x2+1)
To find the derivative of h(x), we can use the chain rule. First, we identify the outer function f(x) as sin(x), and the inner function g(x) as x^2 + 1. Then, we find the derivative of the outer function f'(x) as cos(x), and the derivative of the inner function g'(x) as 2x. Finally, we substitute the functions into the chain rule formula to obtain:
h′(x)=cos(x2+1)⋅2x
Therefore, the derivative of h(x) is given by h'(x) = 2x cos(x^2 + 1).
Conclusion
In conclusion, the derivative of a composite function can be found using the chain rule, which involves finding the derivative of the outer function and the derivative of the inner function, and then multiplying them together. The chain rule is a powerful tool that helps us to find the rate of change of complex functions, and is an essential concept in calculus.