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Chapter 3: Problem 33

Find the derivative of the following functions by first simplifying the expression. \(y=\frac{x-a}{\sqrt{x}-\sqrt{a}} ; a\) is a positive constant.

Short Answer

Expert verified
Question: Find the derivative of the function \(y = \frac{x-a}{\sqrt{x}-\sqrt{a}}\) with respect to x after simplifying the function. Answer: \(\frac{dy}{dx} = \frac{1}{2\sqrt{x}}\)

Step by step solution

01

Simplify the Function

First, we will simplify the function. To do this, notice that we can multiply both the numerator and the denominator by the conjugate of the denominator (i.e., add or subtract the square root expressions): \(y = \frac{x-a}{\sqrt{x}-\sqrt{a}} \times \frac{\sqrt{x} + \sqrt{a}}{\sqrt{x} + \sqrt{a}}\) Upon multiplying both, we get: \(y = \frac{(x-a)(\sqrt{x} + \sqrt{a})}{(\sqrt{x} - \sqrt{a})(\sqrt{x} + \sqrt{a})}\) Now, use the difference of squares formula for the denominator: \(y = \frac{(x-a)(\sqrt{x} + \sqrt{a})}{x-a}\) Now, we can cancel out the (x-a) terms: \(y = \sqrt{x} + \sqrt{a}\) Now that we have a simplified expression, we can proceed to find the derivative.
02

Find the Derivative

Now, let's find the derivative of the simplified function with respect to x: \(\frac{dy}{dx} = \frac{d}{dx}(\sqrt{x} + \sqrt{a})\) Notice that since \(\sqrt{a}\) is a constant, its derivative with respect to x is zero. For \(\sqrt{x}\), we can rewrite it as \(x^{\frac{1}{2}}\): \(\frac{dy}{dx} = \frac{d}{dx}(x^{\frac{1}{2}})\) Now we can use the power rule for differentiation: \(\frac{dy}{dx} = \frac{1}{2}x^{-\frac{1}{2}}\) So, the derivative of the simplified function with respect to x is: \(\frac{dy}{dx} = \frac{1}{2\sqrt{x}}\)

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