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? $$f(x)=x^{3}-4 x, x=2$$

Short answer

The numerical derivative of the function \(f(x)=x^{3}-4x\) at the point \(x=2\) is the result obtained from evaluating \((f(2.001) - f(2))/0.001\). Yes, the function is differentiable at the point \(x=2\).

Step-by-step solution

Write down the function

Start by writing down the function, which is \(f(x)=x^{3}-4x\), and the point where we'll evaluate the derivative, which is \(x=2\).

Calculate \(f(x+h)\)

Using the value of \(h=0.001\), substitute it into the function to calculate \(f(x+h)\). Plug in \(x+h=2+0.001\) into the function to get \(f(2.001)=(2.001)^3-4*(2.001)\).

Evaluate \(f(x + h) - f(x)\)

Now, evaluate \(f(x + h) - f(x)\), which is \((f(2.001) - f(2))\), and simplify the expression.

Evaluate the derivative

Substitute the value of \(h\) and the result from the previous step into the difference quotient formula \(f'(x) = (f(x + h) - f(x))/h\), to get the numerical derivative \((f(2.001) - f(2))/0.001\). Evaluate this division to get the numerical derivative at \(x=2\).

Determine the differentiability

Since the derivative exists at \(x=2\), we conclude that the function is differentiable at this point. Note that a function is differentiable at a point if the derivative exists at that point.