Question

\(f(x)=x^{3}-x, \quad a=1, \quad d x=0.1\)

Short answer

The differential of the function \(f(x) = x^{3} - x\) at \(x = 1\) and \(dx = 0.1\) is 0.2.

Step-by-step solution

Calculate the derivative of the function

First, calculate the derivative of the function \(f(x) = x^3 - x\). The derivative of the function can be computed using basic calculus rules, and is \(f'(x) = 3x^{2} - 1\).

Compute f'(a)

The next step is to compute \(f'(a)\). Here, \(a = 1\), thus \(f'(1) = 3(1)^{2} - 1 = 3 - 1 = 2\).

Determine the differential

One can find the differential of a function \(df(x)\) at a point \(x = a\) using the formula \(df(a) = f'(a) dx\). With \(f'(1) = 2\) and \(dx = 0.1\), we have \(df(1) = 2 * 0.1 = 0.2\).