Question

The area of a square with sides of length \(s\) is given by \(A=s^{2} .\) Find the rate of change of the area with respect to \(s\) when \(s=4\) meters.

Short answer

The rate of change of the area with respect to the side length when the side length is 4 meters is \(8 \) square meters per meter.

Step-by-step solution

Define the Function

The function given, which represents the area of the square in terms of the side length, is \(A(s) = s^{2}\).

Differentiate the Function

The derivative of the function \(A(s)\) with respect to \(s\) is denoted as \(A'(s)\). By applying the power rule for differentiation (which says that the derivative of \(s^n\) is \(n \cdot s^{n-1}\) ), we have \(A'(s) = 2 \cdot s^{1} = 2s\).

Evaluate the Derivative at \(s=4\)

Substitute \(s=4\) into the expression for \(A'(s)\) to find the rate of change of the area at \(s=4\). So, \(A'(4) = 2 \cdot 4 = 8 \) square meters per meter.