Question

In Exercises \(17-26\) , find the numerical derivative of the given function at the indicated point. Use \(h=0.001 .\) Is the function differentiable at the indicated point? $$f(x)=x^{2 / 5}, x=0$$

Short answer

The numerical derivative of the function \(f(x) = x^{2/5}\) at \(x=0\) is \(1/\sqrt[5]{1000}\). The function is differentiable at the point \(x=0\).

Step-by-step solution

Understand the Concept of Numerical Derivative

The numerical derivative of a function at a point can be estimated by taking the differences in y, divided by the differences in x, as x approaches a specific value. The formula for the numerical derivative of a function at a point \(h=0.001\) can be described as: \[ f'(x) \approx \frac {f(x+h) - f(x)}{h} \]

Substitute the Function and x Value

Substitute \(f(x) = x^{2/5}\) and \(x=0\) into the formula: \[ f'(0) \approx \frac {f(0+0.001) - f(0)}{0.001} \]

Solve the Numerical Derivative

Perform the calculation: \[ f'(0) \approx \frac {(0+0.001)^{2/5} - 0^{2/5}}{0.001} = 0.001^{2/5}/0.001= \frac {1}{\sqrt[5]{1000}} \]

Differentiability Test

A function is differentiable at a point if the function is defined there and the derivative exists at that exact point. Since the function \(f(x)=x^{2/5}\) is defined and the derivative exists at \(x=0\), therefore the function is differentiable at \(x=0\).