Question

What is the instantaneous rate of change of y = √x at x = 2?

Answer 1

We are given the function
`y=√(x)` and are asked to find the instantaneous rate of change (slope) at
`x=2`.

First we need to take the derivative of
`y`.

`y=√(x) \Longrightarrow y=x^{(1)/(2) }`

Using the power rule:
`(d)/(dx)[x^(n) ]=nx^(n-1)`

`\Longrightarrow (dy)/(dx) =((1)/(2) )x^{(1)/(2) -1 }`

`\Longrightarrow (dy)/(dx) =(1)/(2) x^{-(1)/(2)}`

`\Longrightarrow (dy)/(dx) =\frac{1}{2x^{(1)/(2)}}`

`\Longrightarrow (dy)/(dx) = y'=(1)/(2√(x))`

Now evaluate
`y'` at
`x=2`.

`\Longrightarrow y'(2)=(1)/(2√(2))`

Thus, the instantaneous rate of change at
`x=2` is
`(1)/(2√(2))`. The second option is correct.