Differentiating and Evaluating Functions: A Comprehensive Guide

In this article, we will delve into the process of differentiating and evaluating functions, specifically focusing on how to find expressions for functions and their derivatives. We will use specific examples to illustrate the methods and ensure clarity in understanding the underlying concepts.

Introduction to Differentiation and Function Evaluation

When dealing with functions in calculus, one of the primary tasks is to find their derivatives. This article covers how to differentiate and evaluate a specific set of functions, namely ( fx ) and ( gx ), where ( fx gx ) and ( gx fx^2 ). We will explore step-by-step methods to express ( fx^2 ) in terms of the given functions.

Step-by-Step Differentiation Process

Step 1: Differentiating the Initial Function

Given that ( fx gx ), we start by differentiating both sides with respect to ( x ). This step involves applying the basic rules of differentiation to both functions.

Step 2: Substituting the Second Function

Since we know that ( gx fx^2 ), we substitute this expression back into the first differentiated equation. This substitution allows us to express ( fx ) in terms of itself squared.

Step 3: Evaluating the Final Expression

The final step is to evaluate the expression for ( fx^2 ). By following the substitutions and simplifications, we can express ( fx^2 ) in a simplified form.

Based on the above steps, the final result is:

Conclusion: Therefore, the final result is:

( fx^2 fx^4 )

Second Order Derivative

For the second-order derivative, we apply the chain rule. Starting with ( fx gx ) and differentiating both sides, we substitute ( gx fx^2 ) into the expression. Then, we use the chain rule to find the second derivative of ( fx^2 ) with respect to ( x ).

Using the Chain Rule for Differentiation

Let's apply the chain rule iteratively to find the second-order derivative. We use substitution and the product rule to achieve this. The final expression is:

( frac{text{d}^2}{text{d}x^2} fx^2 2 cdot fx^2 - 4x^2 cdot fx^2 )

This result is achieved by differentiating and applying the chain rule at each step.

Conclusion

In conclusion, this article has provided a detailed explanation of how to differentiate and evaluate functions, emphasizing the use of the chain rule and substitution. By following the outlined steps, you can effectively find expressions for complex functions and their derivatives.

For further information and practice, consider exploring additional examples of function differentiation and applying similar techniques to other mathematical problems.