Final answer:
Differentiating the function g(x) using the quotient rule involves taking the derivatives of the numerator and denominator and substituting into the formula for the derivative of a quotient.
Step-by-step explanation:
To differentiate the function g(x) = x²+5/(x²+6x), we need to apply the quotient rule. The quotient rule states that if we have a function f(x)/g(x), its derivative will be (f'(x)g(x) - f(x)g'(x))/(g(x))^2. In this case, our f(x) is x²+5 and g(x) is x²+6x.
First, we find the derivative of f(x), which is 2x. Then we find the derivative of g(x), which is 2x+6. Applying the quotient rule gives us:
d/dx[g(x)] = ( (2x)(x²+6x) - (x²+5)(2x+6) ) / (x²+6x)²
The student can leave the answer in this form, without simplifying further as per the instructions.