Find the derivative of the function and evaluate
f '(x)
at the given value of x.
f(x) = 4x + 5/4x − 5 ; x = 4
To find the derivative of the function f(x), we can use the quotient rule. The quotient rule states that if we have a function f(x) = g(x) / h(x), where g(x) and h(x) are differentiable functions, then the derivative f'(x) is given by:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
In this case, g(x) = 4x + 5 and h(x) = 4x - 5. Let's find the derivatives of g(x) and h(x) first.
The derivative of g(x) = 4x + 5 is obtained by taking the derivative of each term:
g'(x) = 4
The derivative of h(x) = 4x - 5 is also obtained by taking the derivative of each term:
h'(x) = 4
Now, we can plug these values into the quotient rule formula to find f'(x):
f'(x) = (4 * (4x - 5) - (4x + 5) * 4) / ((4x - 5)^2)
Simplifying this expression, we get:
f'(x) = (16x - 20 - 16x - 20) / (16x^2 - 40x - 20x + 25)
f'(x) = (-40) / (16x^2 - 60x + 25)
To evaluate f'(x) at x = 4, we substitute x = 4 into the expression we obtained:
f'(4) = (-40) / (16(4)^2 - 60(4) + 25)
f'(4) = (-40) / (256 - 240 + 25)
f'(4) = (-40) / (41)
The value of f'(x) at x = 4 is -40/41.