Let f(7)=0, f'(7)=14, g(7)=1, g'(7)=1/7. Find h'(7) if h(x)=f(x)/g(x)
Just use the quotient rule:
h' = (f'g - fg')/g^2
= (14*1 - 0*1/7)/1^2 = 14
To find h'(7), we can use the quotient rule, which states that if we have a function h(x) = f(x) / g(x), then the derivative of h(x) is given by the formula:
h'(x) = (f'(x) * g(x) - f(x) * g'(x)) / [g(x)]^2
Given that f(7) = 0, f'(7) = 14, g(7) = 1, and g'(7) = 1/7, we can substitute these values into the formula:
h'(7) = (f'(7) * g(7) - f(7) * g'(7)) / [g(7)]^2
= (14 * 1 - 0 * (1/7)) / [1]^2
= 14 / 1
= 14
Therefore, h'(7) = 14.
To find h'(7), we need to calculate the derivative of the function h(x) = f(x)/g(x) and then evaluate it at x = 7.
Step 1: Find the derivative of h(x) = f(x)/g(x)
To find the derivative of a quotient of two functions, we can use the quotient rule.
The quotient rule states that if we have two functions u(x) and v(x), then the derivative of their quotient is given by:
(h(x))' = [u'(x) * v(x) - u(x) * v'(x)] / [v(x)]^2
In this case, u(x) = f(x) and v(x) = g(x), so we can substitute these values into the quotient rule.
(h(x))' = [f'(x) * g(x) - f(x) * g'(x)] / [g(x)]^2
Step 2: Evaluate h'(7)
Now that we have the derivative function h'(x), we can evaluate it at x = 7 to find h'(7).
Substitute x = 7 into the derivative function:
h'(7) = [f'(7) * g(7) - f(7) * g'(7)] / [g(7)]^2
Substitute the given values:
h'(7) = [14 * 1 - 0 * (1/7)] / 1^2
= 14 / 1
= 14
Therefore, h'(7) = 14.