Question

What is the derivative of `y=tan(x)/x`?

Answer 1

This function, in the form of `y = f(x) = g(x)/(h(x))`, is a perfect candidate for using the quotient rule.

The quotient rule states that the derivative of `y` with respect to `x` can be solved with the following formula:

Quotient rule: `y'= f'(x) = (g'(x)h(x) - g(x)h'(x))/(h(x)^2)`

In this problem, we can assign the following values to the variables in the quotient rule:

`g(x) = tan(x)`
`h(x) = x`

`g'(x) = sec^2(x)`
`h'(x) = 1`

If we plug these values into the quotient rule, we get the final answer:

`y' = (sec^2(x) * x - tan(x) * 1)/x^2 = (xsec^2(x) - tan(x))/x^2`