Final answer:
The second derivative of the function y = (x - 4)/(x + 5) is found to be -20/(x + 5)^3 by first finding the first derivative using the quotient rule and then differentiating this result.
Step-by-step explanation:
To find the second derivative of the function y = (x - 4)/(x + 5), we first need to find its first derivative. Using the quotient rule for differentiation, which states that d/dx (u/v) = (v(u') - u(v'))/v^2 where u' and v' are the derivatives of u and v respectively, we can differentiate the function.
The first derivative, which we will call y', is:
y' = d/dx [(x - 4)/(x + 5)] = ((x + 5)(1) - (x - 4)(1))/(x + 5)^2 = (1 + 9)/(x + 5)^2 = 10/(x + 5)^2
Next, we differentiate the first derivative to find the second derivative, y'':
y'' = d/dx [10/(x + 5)^2] = 10 d/dx [(x + 5)^{-2}]
Using the power rule combined with the chain rule, we get:
y'' = 10 * -2(x + 5)^{-3} * 1 = -20/(x + 5)^3
Therefore, the second derivative of the given function is -20/(x + 5)^3.