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Chapter 2: Problem 65

Use a graphing utility to find the \(x\) -values at which \(f\) is differentiable. \(f(x)=x^{2 / 5}\)

Short Answer

Expert verified
The function \(f(x)=x^{2 / 5}\) is differentiable for all real \(x\) except \(x = 0\).

Step by step solution

01

Differentiation

To begin, differentiate \(f(x)=x^{2/5}\) using the power rule of differentiation. The power rule states that if a function is in the form \(x^n\), its derivative is \(nx^{n-1}\).\nSo, the derivative \(f'(x)\) of \(f(x)\) is \(2/5\) * \(x^{(2/5 - 1)}\) which is \(2/5\) * \(x^{-3/5}\).
02

Examine the Derived Function

The derived function \(f'(x)\) = \(2/5\) * \(x^{-3/5}\) exists for all real x except for x = 0. This can be deduced by understanding that the derivative function has a negative power of x, hence it becomes undefined when x = 0.
03

Conclusion

Given that the derivative exists for all real numbers except for 0, it can be concluded that the function \(f(x)=x^{2/5}\) is differentiable for all real numbers except at x = 0. Remember, a function is differentiable at a point if the derivative exists at that point. So, any x-values can be put into the function except for 0.

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Most popular questions from this chapter

Existence of an Inverse Determine the values of \(k\) such that the function \(f(x)=k x+\sin x\) has an inverse function.

Find the second derivative of the function. \(g(x)=\sqrt{x}+e^{x} \ln x\)

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Prove (Theorem 2.3) that \(\frac{d}{d x}\left[x^{n}\right]=n x^{n-1}\) for the case in which \(n\) is a rational number. (Hint: Write \(y=x^{p / q}\) in the form \(y^{q}=x^{p}\) and differentiate implicitly. Assume that \(p\) and \(q\) are integers, where \(q>0 .)\)

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