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

Question

asked Nov 1, 2024 144k views

1 Answer

James Law · Answer 1 · 2024-11-06T17:24:47+0000

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
.

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

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
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.