Find the derivative of the function.
f(x) = text(ln)(9 x^2 - 3 x + 1)
To find the derivative of the function f(x) = ln(9x^2 - 3x + 1), we can use the chain rule of differentiation. The chain rule states that for a composite function f(g(x)), the derivative is given by f'(g(x)) * g'(x).
In this case, the outer function is f(x) = ln(x) and the inner function is g(x) = 9x^2 - 3x + 1.
Step 1: Find the derivative of the outer function.
The derivative of ln(x) with respect to x is 1/x. So, the derivative of the outer function is f'(x) = 1/(9x^2 - 3x + 1).
Step 2: Find the derivative of the inner function.
To find the derivative of g(x) = 9x^2 - 3x + 1, we apply the power rule and the sum rule. The power rule states that the derivative of x^n is n*x^(n-1), and the sum rule states that the derivative of the sum of two functions is the sum of their derivatives.
Applying the power rule, we get:
g'(x) = d/dx (9x^2) - d/dx (3x) + d/dx (1)
= 18x - 3
Step 3: Multiply the derivatives found in Step 1 and Step 2.
Using the chain rule, we multiply the derivative of the outer function (1/(9x^2 - 3x + 1)) with the derivative of the inner function (18x - 3):
f'(x) = (1/(9x^2 - 3x + 1)) * (18x - 3)
Therefore, the derivative of f(x) = ln(9x^2 - 3x + 1) is f'(x) = (18x - 3)/(9x^2 - 3x + 1).