Find the derivative of the function f by using the rules of differentiation.
f(x)=(9x^3+3x^2+10x-6)/(x)
The rule is:
the denominator times the derivative of the numerator - the numerator times the derivative of the denominator ALL OVER the denominator squared.
To find the derivative of the function f(x) = (9x^3 + 3x^2 + 10x - 6) / x, we can use the quotient rule for differentiation. The quotient rule states that if we have a function u(x) / v(x), where u(x) and v(x) are differentiable functions, then the derivative of the function is given by:
f'(x) = (v(x) * u'(x) - u(x) * v'(x)) / (v(x))^2
In this case, u(x) = 9x^3 + 3x^2 + 10x - 6 and v(x) = x. Let's find the derivatives of u(x) and v(x) first:
u'(x) = d/dx (9x^3 + 3x^2 + 10x - 6) = 27x^2 + 6x + 10
v'(x) = d/dx (x) = 1
Now, we can substitute these values into the quotient rule formula to find the derivative:
f'(x) = ((x)(27x^2 + 6x + 10) - (9x^3 + 3x^2 + 10x - 6)(1)) / (x)^2
Simplifying this expression gives us the derivative of f(x):
f'(x) = (27x^3 + 6x^2 + 10x^2 - 9x^3 - 3x^2 - 10x + 6) / x^2
Combining like terms and simplifying further:
f'(x) = (18x^2 - 4x + 6) / x^2
So, the derivative of f(x) is (18x^2 - 4x + 6) / x^2.