How to Find dy/dx when y 1x / (1-x^(1/2))

Understanding how to find the derivative of a function, especially one involving square roots, is a fundamental concept in calculus. In this article, we will walk through the process of finding dy/dx for the function y 1x / (1 - x^(1/2)). Let's break it down step-by-step using various mathematical techniques and rules.

Step 1: Substitution

First, let's set u 1x / (1 - x^(1/2)). This simplifies our problem by breaking it down into more manageable parts. We will use the chain rule to find the derivative of y u with respect to x

The chain rule states that dy/dx dy/du * du/dx. Given y 1/(2*sqrt(u)) * u, we need to find dy/du

dy/du 1/(2 * sqrt(u)) - u/2 * 1/(2 * sqrt(u)^3) When simplified, this becomes dy/du 1/(2*sqrt(u)) - 1/(4 * u * sqrt(u)) 1/(4 * u * sqrt(u))

Step 2: Simplifying u

Now, we need to find du/dx where u 2/(1-x^(1/2)) - 1

First, let's simplify the expression for u

u 2/(1-x^(1/2)) - 1

Now, differentiate u with respect to x

du/dx -2/(1-x^(1/2))^2 * -1/2 * x^(-1/2) Simplifying, we get du/dx 1/x^(1/2)/(1-x^(1/2))^2

Step 3: Combining the Results

Now, we can combine dy/du and du/dx to find dy/dx

dy/dx dy/du * du/dx (1/4 * u * sqrt(u)) * (1/x^(1/2)/(1-x^(1/2))^2)

Substituting back the expression for u (2/(1-x^(1/2)) - 1), we get

y 1/(2*sqrt(2/(1-x^(1/2)) - 1)) * (1/x^(1/2)/(1-x^(1/2))^2)

Alternative Approach Using the Quotient Rule

Another method to find the derivative of y √(1x / (1 - x^(1/2))) is to use the quotient rule. The quotient rule states that dy/dx (g * d/dx(f) - f * d/dx(g)) / g^2

Let's define f(x) 1 - x^(1/2) and g(x) 1x

dy/dx [1*x^(-1/2) - 1/2 * (1 - x^(1/2))^(-1/2)] / [1 - x^(1/2)]^2 * x

Further simplifying, we get:

dy/dx (1/x^(1/2) - 1/2 / sqrt(1 - x^(1/2))) / [1 - x^(1/2)]^2 * x

dy/dx (2 - 1/x^(1/2) * (1 - x^(1/2))) / [2 * x^(1/2) * (1 - x^(1/2))^2]

dy/dx (2 - 2 * sqrt(1 - x^(1/2))) / [2 * x^(1/2) * (1 - x^(1/2))^2]

Simplifying further:

dy/dx 1 / (1 - sqrt(1 - x^(1/2)) * 1/x^(1/2))

Final Answer

The derivative of the function y 1x / (1 - x^(1/2)), after following the steps and simplifications, is:

dy/dx 1 / (1 - x * sqrt(1 - x^(1/2)))

This result is more intricate and requires a careful approach to handle the square roots and algebraic manipulations involved.

Conclusion

Mastering the techniques of finding derivatives, especially those involving square roots and more complex algebraic expressions, is crucial for advanced calculus. This process not only enhances your problem-solving skills but also deepens your understanding of calculus principles.

Key Takeaways

Substitution and the chain rule can simplify the derivative calculation. The quotient rule provides an alternative approach for complex functions. Algebraic manipulation is key to simplifying the final form of the derivative.

Keywords

derivative, square root, calculus