Question

Find the instantaneous rate of change of g(x) = x at x = 6.

Answer 1

The instantaneous rate of change of
`\(g(x) = x\)` at
`\(x = 6\)` is 1.

Step-by-step explanation:

The instantaneous rate of change at a specific point is given by the derivative of the function at that point. For
`\(g(x) = x\)` , the derivative
`\(g'(x)\)` is equal to 1, as the derivative of
`\(x\)` with respect to
`\(x\)` is 1. Therefore, at any point
`\(x\),` the instantaneous rate of change of
`\(g(x)\)` is 1.

To find the instantaneous rate of change at
`\(x = 6\),` we evaluate
`\(g'(x)\) at \(x = 6\).` Substituting
`\(x = 6\) into \(g'(x) = 1\) gives us \(g'(6) = 1\).` Thus, the instantaneous rate of change of
`\(g(x) = x\) at \(x = 6\` ) is 1.

Understanding the concept of instantaneous rate of change is crucial in calculus. It represents the rate at which a function is changing at a specific point and is often interpreted as the slope of the tangent line to the graph of the function at that point.

In this case, for the simple linear function
`\(g(x) = x\),` the instantaneous rate of change is a constant 1, indicating a constant slope of the tangent line across all points on the graph.