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? $$f(x)=x^{4 / 5}, x=0$$

Short answer

The numerical derivative of the function \(f(x) = x^{4 / 5}\) at \(x = 0\) is \(f'(0) = 0.63096\). The function is differentiable at \(x = 0\).

Step-by-step solution

Calculating the function at \(x\) and \(x + h\)

To calculate the numerical derivative, we first need to calculate the function at the point \(x\) and \(x + h\). Given \(x = 0\) and \(h = 0.001\), we substitute these values into the function \(f(x) = x^{4/5}\) to get \(f(x) = 0\) and \(f(x + h) = 0.001^{4/5}\).

Plugging the values into the numerical derivative formula

We then substitute the calculated values into the formula for the numerical derivative, which is \(f'(x) = \frac{f(x + h) - f(x)}{h}\). Doing this, we get \(f'(0) = \frac{0.001^{4/5} - 0}{0.001}\).

Computing the numerical derivative

After calculating the above equation, we get \(f'(0) = 0.63096\).

Checking differentiability at the point

As we have successfully calculated the derivative at the point without encountering any undefined or infinite value issues, the function \(f(x) = x^{4/5}\) is indeed differentiable at \(x = 0\).