Use the following function values to calculate H '(1).

Question

f (1) f '(1) g(1) g'(1)
3 -4 8 -1

H(x) = x/(g(x) f(x))

How would this problem be done...
i found H(x) to be 1/24 but don't know how to find the derivative of h'(1)

Answer 1

Use the quotient rule for H(x).

It looks like you have forgotten the parentheses between f(x)g(x) to give
H(x)=x/(g(x)f(x))
where the denominator contains the product of f(x) and g(x).
When transcribing equations, parentheses are required on the numerator and denominator if they consist of more than one term.

Find the derivative of H(x) using the quotient rule:
d(f(x)/g(x))/dx = f'(x)g(x)-f(x)g'(x)/g(x)²
and the product rule:
d(f(x)g(x))/dx = f(x)g'(x)+f'(x)g(x)

For
H(x)=x/(g(x)f(x))
we get
H'(x)
=(g(x)f(x)*1 - x(f(x)g(x))')/(f(x)g(x))²
=(g(x)f(x)-x(f(x)g'(x)+f'(x)g(x)))/(f(x)g(x))²

All of the quantities are known on the right-hand-side and can be evaluated numerically.