Use the quotient rule to find the derivative of the following function.𝑓(𝑥)=ln(𝑥)𝑥2

The quotient rule states that the derivative of a quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all over the square of the denominator.

The function f(x) = ln(x)/x^2 is a quotient of two functions, u(x) = ln(x) and v(x) = x^2.

First, we need to find the derivatives of u(x) and v(x).

The derivative of u(x) = ln(x) is u'(x) = 1/x.

The derivative of v(x) = x^2 is v'(x) = 2x.

Now we can apply the quotient rule:

f'(x) = (v(x) * u'(x) - u(x) * v'(x)) / (v(x))^2 = (x^2 * (1/x) - ln(x) * 2x) / (x^2)^2 = (x - 2x ln(x)) / x^4 = x(1 - 2 ln(x)) / x^4

We can simplify this by canceling out an x from the numerator and denominator:

f'(x) = (1 - 2 ln(x)) / x^3

So, the derivative of the function f(x) = ln(x)/x^2 is f'(x) = (1 - 2