Final answer:
The first derivative of f(x) = x / (4x + 5) is f'(x) = 5/(4x + 5)^2, and the second derivative is f''(x) = -40/(4x + 5)^3.
Step-by-step explanation:
The question asks for the first and second derivatives of the function f(x) = x / (4x + 5). To find these, we will use the quotient rule for derivatives. The quotient rule states that for a function of the form g(x)/h(x), the derivative, denoted by g'(x), is given by (g'(x)h(x) - g(x)h'(x))/(h(x))^2.
Applying the quotient rule:
- f'(x) = (1(4x + 5) - x(4))/(4x + 5)^2
- f'(x) = (4x + 5 - 4x)/(4x + 5)^2
- f'(x) = 5/(4x + 5)^2
Now, for the second derivative, f''(x):
- f''(x) = d/dx [5/(4x + 5)^2]
- f''(x) = -10(4)/(4x + 5)^3 [applying the chain rule and the power rule]
- f''(x) = -40/(4x + 5)^3
Therefore, the first derivative of the function is f'(x) = 5/(4x + 5)^2 and the second derivative is f''(x) = -40/(4x + 5)^3.