Find the derivative of f. you may simplify before differentiating
f(x)= x2-7x+12/x4-8x3+13x2+30x-72
Surely you meant ...
f(x) = (x^2-7x+12)/(x^4-8x^3+13x^2+30x-72)
= (x-3)(x-4)/(x^4-8x^3+13x^2+30x-72)
I checked by synthetic division if those factors are contained in the denominator, and they are, so
= 1/(x^2 - x - 6)
= (x^2 - x - 6)^-1
f '(x) = -(x^2 - x - 6)^-2 (2x-1)
= (1 - 2x)/(x^2 - x - 6)^2
To find the derivative of f(x), you can use the quotient rule. The quotient rule states that if you have a function of the form h(x) = g(x) / p(x), where g(x) and p(x) are both functions, then the derivative of h(x) can be found by using the formula:
h'(x) = (g'(x) * p(x) - g(x) * p'(x)) / (p(x))^2
In this case, f(x) = (x^2 - 7x + 12) / (x^4 - 8x^3 + 13x^2 + 30x - 72. Let's label g(x) = x^2 - 7x + 12 and p(x) = x^4 - 8x^3 + 13x^2 + 30x - 72.
Now, let's find the derivatives of g(x) and p(x).
g'(x) = 2x - 7 (using the power rule for derivatives on each term)
p'(x) = 4x^3 - 24x^2 + 26x + 30 (using the power rule for derivatives on each term)
Now, substitute these values into the quotient rule formula:
f'(x) = [(2x - 7) * (x^4 - 8x^3 + 13x^2 + 30x - 72) - (x^2 - 7x + 12) * (4x^3 - 24x^2 + 26x + 30)] / (x^4 - 8x^3 + 13x^2 + 30x - 72)^2
Simplifying this expression further may be difficult, so it's recommended to keep the derivative in factored form like this.