Problem 54 - College Algebra and Calculus: An Applied Approach

Question

Describe the \(x\) -values at which the function is differentiable. Explain your reasoning. $$ y=x^{2 / 5} $$

Step-by-Step Solution

Answer

The function \(y=x^{2 / 5}\) is differentiable for all real values of \(x\).

Step 1: Understand the Definitions

Recall that a function \(f(x)\) is differentiable at a particular \(x\) -value if the function's derivative exists at that \(x\) -value. Essentially, this means that the function has a defined slope at that point.

Step 2: Differentiate the Function

We need to calculate the derivative of the function. In this case, the function \(y=x^{2 / 5}\) can be differentiated using the power rule, which states that the derivative of \(x^n\) is \(nx^{n?1}\). Applying this rule to the function, we compute the derivative as \(y'=(2 / 5)x^{-3 / 5}\).

Step 3: Determine Where the Derivative Is Undefined

A function is only differentiable where its derivative is defined. For the derivative function \(y'=(2 / 5)x^{-3 / 5}\), it is undefined when the denominator is zero. Since there are no values of \(x\) that will make the denominator zero, the derivative is defined for all real values of \(x\).