Estimate the instantaneous rate of change at the point indicated. (Round your answer to four decimal places.)

y(x) =
1/ x + 2; x = 2

To estimate the instantaneous rate of change at a specific point, we can calculate the derivative of the function. In this case, we need to find the derivative of the function y(x) = 1/x + 2.

First, let's rewrite the function in a more simplified form:

y(x) = 1/x + 2 = x^(-1) + 2.

The derivative of a function f(x) is calculated by taking the derivative of each term separately.

The derivative of x^(-1) can be found using the power rule. According to the power rule, the derivative of x^n is n*x^(n-1). In this case, n is -1:

d/dx (x^(-1)) = -1*x^(-1-1) = -1*x^(-2) = -x^(-2).

Now, since the derivative of a constant (2 in this case) is zero, the derivative of the term 2 is simply 0.

Therefore, the derivative of y(x) = x^(-1) + 2 is:

dy/dx = -x^(-2).

To estimate the instantaneous rate of change at x = 2, we substitute the value into the derivative function:

dy/dx = -2^(-2) = -1/4 = -0.25.

Hence, the instantaneous rate of change at the point (x = 2) for the given function is approximately -0.25.

To find the instantaneous rate of change at a specific point, we need to find the derivative of the function at that point.

Given the function:
y(x) = 1/x + 2

To find the derivative of this function, we can use the power rule for differentiation.

Power rule for differentiation: d/dx (x^n) = n*x^(n-1)

Let's apply the power rule to find the derivative of y(x):
y'(x) = d/dx (1/x + 2)
= d/dx (x^(-1) + 2)
= -1*x^(-1-1) + 0
= -1/x^2

Now, let's find the instantaneous rate of change at x = 2 by substituting this value into the derivative we just found:

y'(2) = -1/2^2
= -1/4
≈ -0.2500

Therefore, the estimated instantaneous rate of change at x = 2 is approximately -0.2500.

I assume you want dy/dx

y = 1/x + 2
dy/dx = -1/x^2
when x = 2
dy/dx = -1/4