Calculate the derivative of the function. Then find the value of the derivative as specified.
f(x) = 8/(x+2); f '(0)
f'(x) = -8/(x+2)^2
f'(0) = -8/4 = -2
To calculate the derivative of the function f(x) = 8/(x+2), we can use the quotient rule. The quotient rule states that if you have a function of the form f(x) = u(x)/v(x), where u(x) and v(x) are both differentiable functions, then the derivative of f(x) is given by:
f'(x) = (u'(x)v(x) - u(x)v'(x))/[v(x)]^2
In this case, u(x) = 8 and v(x) = (x+2). Let's calculate the derivative using the quotient rule:
First, we need to find the derivative of u(x) and v(x).
u'(x) = 0 (since 8 is a constant, its derivative is 0)
v'(x) = 1 (the derivative of x+2 is 1)
Now we can substitute these values into the quotient rule formula:
f'(x) = (0 * (x+2) - 8 * 1)/[(x+2)]^2
= -8/[(x+2)]^2
To find the value of the derivative at x = 0, we substitute x = 0 into the expression we obtained for f'(x):
f'(0) = -8/[0+2]^2
= -8/4
= -2
So, the value of the derivative f'(0) is -2.