use the quotient rule to differentiate the following function f(x)=(1+In(x))/(x)
dy/dx = [1/x - (1+ln(x)]/ x^2
Sorry I missed a ')'
dy/dx = [1/x - (1+ln(x))]/ x^2
To differentiate the function f(x) = (1 + ln(x))/x using the quotient rule, follow these steps:
1. Begin by identifying the two parts of the function: the numerator and the denominator.
In this case, the numerator is 1 + ln(x) and the denominator is x.
2. Apply the quotient rule, which states that the derivative of a quotient is given by:
(denominator * derivative of numerator - numerator * derivative of denominator) / (denominator^2)
3. Let's differentiate the numerator and denominator separately.
a. Derivative of the numerator:
The numerator is 1 + ln(x). To differentiate this, we differentiate each part separately.
The derivative of 1 is 0 (since it's a constant).
The derivative of ln(x) is 1/x (using the derivative of natural logarithm rule).
b. Derivative of the denominator:
The denominator is x, which is a simple polynomial.
The derivative of x is 1 (using the derivative of a power rule).
4. Plug the results from steps 3a and 3b into the quotient rule formula:
[(x * derivative of numerator) - (numerator * derivative of denominator)] / (x^2)
= [(x * (0 + 1/x)) - ((1 + ln(x)) * 1)] / (x^2)
= (x/x) - (1 + ln(x)) / (x^2)
= 1 - (1 + ln(x)) / (x^2)
= 1 - (1/x + ln(x))/x^2
Therefore, the derivative of f(x) = (1 + ln(x))/x is 1 - (1/x + ln(x))/x^2.