Question

Compute the instantaneous rate of change of the function at the indicated x-value. f(x) = 9√x; x = 4

Answer 1

So, at
`\(x = 4\)`, the function
`\(f(x) = 9√(x)\)`is changing at a rate of
`\((9)/(4)\)`, which means it's increasing or decreasing at that rate at that particular point.

Explanation:

Certainly! To find the instantaneous rate of change of the function
`\(f(x) = 9√(x)\) at \(x = 4\)`, think of it as how fast the function is changing at that specific point.

1. Start with the function
`\(f(x) = 9√(x)\).`

2. Find the derivative
`\(f'(x)\)` of this function, which tells you how fast it's changing at any
`\(x\).`

3. The derivative
`\(f'(x)\) is \((9)/(2√(x))\).` This formula describes how the rate of change depends on the value of
`\(x\).`

4. Now, plug in the specific
`\(x\)-value, which is 4:`

`\[f'(4) = (9)/(2√(4))\]`

5. Simplify the expression:

`\[f'(4) = (9)/(2 \cdot 2)\]`

`\[f'(4) = (9)/(4)\]`

So, at
`\(x = 4\)`, the function
`\(f(x) = 9√(x)\)`is changing at a rate of
`\((9)/(4)\)`, which means it's increasing or decreasing at that rate at that particular point.