Consider the function f(x) for which f(0)=7 and f'(0)=7. Find h'(0) for th function h(x)= 1/f(x). h'(0)=
To find the derivative of the function h(x) = 1/f(x) at x = 0, we can use the quotient rule.
The quotient rule states that if we have two functions u(x) and v(x), the derivative of their quotient (u/v)' can be calculated as:
(u/v)' = (u'v - uv') / v^2
In this case, u(x) = 1 and v(x) = f(x).
First, let's find u'(x):
u'(x) = 0 (since 1 is a constant)
Next, let's find v'(x), which is the derivative of f(x):
Given f'(0) = 7, we have v'(0) = 7.
Now, let's substitute these values into the quotient rule formula:
(h'(x))|x=0 = (u'v - uv') / v^2
= (0*f(x) - 1*f'(x)) / (f(x))^2
= (-7) / (f(x))^2
Now, we need to evaluate this expression at x = 0:
(h'(0)) = (-7) / (f(0))^2
= (-7) / (7)^2
= (-7) / 49
= -7/49
Therefore, h'(0) = -7/49.