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A composite function is a function that is formed when one function is applied to the result of another function. In the exercise, we understand that we are dealing with a composite function when we see the expression \( \sqrt{2x+1} \). Here, there is an inner function \( g(x) = 2x+1 \) and an outer function \( f(u) = \sqrt{u} \), where \( u = g(x) \).
Recognizing composite functions is critical because they often require special differentiation techniques, like the chain rule, which we will discuss next.
In general, with composite functions, you will see a main function wrapping or encapsulating an inner function. This makes it crucial to identify each part when preparing to differentiate.

The Chain Rule is a fundamental technique in calculus used to differentiate composite functions. This rule is important because it allows us to find the derivative of an outer function with respect to the inner function. Then, we multiply this result by the derivative of the inner function itself.
In simple terms, the Chain Rule helps us "chain" together the derivatives of nested functions.

• The rule states: if you have a composite function \( f(x) = (g(x))^n \), then its derivative is \( f'(x) = n(g(x))^{n-1} \cdot g'(x) \).
• For example, in our problem, \( n = \frac{1}{2} \) as the power of the inner function \( g(x) = 2x+1 \).

This method ensures that we correctly account for changes within both the outer and inner functions.
Focusing on practicing the Chain Rule for various composite functions will help reinforce your understanding and skill in differentiation.

When differentiating a composite function using the Chain Rule, it’s important to follow a structured approach. Let's break down the steps using the exercise example \( \sqrt{2x+1} \):- **Step 1**: **Identify the composite function**, noting the inner and outer components. Here, the inner function is \( g(x) = 2x+1 \) and the outer function in exponential form is \( (g(x))^{1/2} \).- **Step 2**: Rewrite the function in a form that's easier to differentiate. This might involve expressing roots as fractional exponents, i.e., \( \sqrt{2x+1} \) becomes \( (2x+1)^{1/2} \).- **Step 3**: **Differentiate the outer function** with respect to the inner function. This results in \( \frac{1}{2}(2x+1)^{-1/2} \).- **Step 4**: **Differentiate the inner function**, which is simply the derivative of \( g(x) = 2x+1 \), giving \( g'(x) = 2 \).- **Step 5**: Finally, **multiply the derivative of the outer function by the derivative of the inner function**: \( f'(x) = \frac{1}{2}(2x+1)^{-1/2} \cdot 2 \).- **Step 6**: **Simplify the expression**, resulting in \( f'(x) = (2x+1)^{-1/2} \), or as a fraction, \( \frac{1}{\sqrt{2x+1}} \).Following these steps ensures you approach each problem systematically, reducing errors and clarifying the differentiation process.