Problem 27 - College Algebra and Calculus: An Applied Approach

Question

Use Example 6 as a model to find the derivative. $$ y=\frac{\sqrt{x}}{x} $$

Step-by-Step Solution

Answer

The derivative of the function \(y=\frac{\sqrt{x}}{x}\) is \(y' = -\frac{1}{2x^{\frac{3}{2}}}\)

Step 1: Simplify the Function

First, observe that \(y=\frac{\sqrt{x}}{x}\) can be simplified. \(\sqrt{x}\) is same as \(x^{\frac{1}{2}}\), and dividing by x is same as multiplying by \(x^{-1}\). Thus, the function \(y\) can be rewritten as \(y = x^{\frac{1}{2}} \cdot x^{-1} = x^{\frac{1}{2} - 1} = x^{-\frac{1}{2}}\).

Step 2: Apply the Power Rule

Having simplified the function, we can now find its derivative using the power rule for derivatives. The power rule states that the derivative of \(x^n\) is \(n * x^{n - 1}\). In this case, \(n = -\frac{1}{2}\). So applying the Power Rule, the derivative \(y'\) becomes \(-\frac{1}{2} * x^{-\frac{1}{2} - 1} = -\frac{1}{2} * x^{-\frac{3}{2}}\).

Step 3: Simplify the Derivative

Observe that the derivative \(-\frac{1}{2} * x^{-\frac{3}{2}}\) could be left in this form, but it may be written more simply in a different form by realizing \(x^{-\frac{3}{2}}\) could be rewritten as \(\frac{1}{x^{\frac{3}{2}}}\). Doing that simplification, we get \(y' = -\frac{1}{2} * \frac{1}{x^{\frac{3}{2}}}\).