Deriving g(x) from f(x) [x2] / [x-4] and g(x) f(x)2

Welcome to this in-depth exploration of deriving the function g(x) from the given function f(x) using mathematical techniques such as the chain rule and algebraic manipulation. In this article, we will break down the process and the nuances involved.

Defining f(x)

Let's start with the function f(x) frac{x^2}{x-4}. To understand how to derive its components and ultimately g(x), we first need to manipulate and integrate this function.

Integration of f(x)

Integrating f(x) frac{x^2}{x-4} directly can be quite complex. However, we can simplify it for integration. Let's rewrite it in a more manageable form:

fx frac{x^2}{x-4} frac{x^2 - 16 16}{x-4} frac{(x^2 - 16) 16}{x-4} x 4 frac{16}{x-4}

Integrating this, we get:

int fx , dx int (x 4 frac{16}{x-4}) , dx frac{x^2}{2} 4x 16ln|x-4| C

Derivation of g(x)

Given that g(x) f(x)^2, we need to find the expression for g(x). Let's denote u x^2 and g(x) u^2 / (u-4).

Using the chain rule, we can express the derivative g'(x). The chain rule states that g'(x) 2xf'(x). Now, we need to find f'(x).

f'x frac{d}{dx} (frac{x^2}{x-4}) frac{(x^2 - 16) 16}{(x-4)^2} 1 frac{8}{x-4}

Now, applying the chain rule:

g'x 2xf'x 2x left(1 frac{8}{x-4}right) 2x frac{16x}{x-4}

Final Form of g(x)

From the above steps, we can derive the final form of g(x) as:

g(x) frac{2x^3 - 4x}{x^2 - 4}

Verification and Conclusion

To verify the solution, we can substitute back into the original function and derive the expression:

g(x) f(x)^2 left(frac{x^2}{x-4}right)^2 frac{x^4}{(x-4)^2}

Then, using the chain rule, we can derive:

g'x 2xf'x 2x left(1 frac{8}{x-4}right) 2x frac{16x}{x-4}

This matches our earlier derived expression, confirming the correctness of our solution.

Conclusion

In this article, we have shown how to derive the function g(x) f(x)^2 from the given function f(x) frac{x^2}{x-4} using the chain rule and algebraic manipulation. The process involves integration, substitution, and applying the chain rule. Understanding these techniques is crucial for anyone working with complex mathematical functions in fields such as calculus, physics, and engineering.

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