Calculate the derivative indicated.
y '(25), y = �ãx + 2
x
To calculate the derivative of a function, you can apply the power rule for derivatives. The power rule states that for a function of the form f(x) = x^n, the derivative is found by multiplying the exponent by the coefficient and subtracting 1 from the exponent.
In this case, we have the function y(x) = (x + 2) / x. To find the derivative y'(x), we need to differentiate each term separately.
Let's first simplify the function by expanding the numerator:
y(x) = x/x + 2/x
= 1 + 2/x
Now we can differentiate each term:
The derivative of 1 is 0, since it's a constant.
To find the derivative of 2/x, we can rewrite it as 2x^(-1) and apply the power rule.
The power rule states that d/dx(x^n) = nx^(n-1), so applying it to 2x^(-1), we get:
d/dx(2x^(-1)) = -2x^(-1-1)
= -2x^(-2)
= -2/x^2
Now let's combine the derivatives we found:
y'(x) = 0 + (-2/x^2)
= -2/x^2
To find y'(25), we substitute x = 25 into the derived function:
y'(25) = -2/25^2
= -2/625
Therefore, y'(25) = -2/625.