How do you use the quotient rule to differentiate the function f(x) = tan(x) / sec(x)?

To differentiate the function f(x) = tan(x) / sec(x) using the quotient rule, we follow these steps:

The quotient rule states that if you have a function that is the quotient of two other functions, say u(x) / v(x), then the derivative is given by:

f'(x) = (u’v – uv’) / v².

In our case, let:

• u = tan(x),
• v = sec(x).

Next, we need to find the derivatives of u and v:

• u’ = sec²(x) (the derivative of tan(x)),
• v’ = sec(x)tan(x) (the derivative of sec(x)).

Now we can substitute these derivatives back into the quotient rule formula:

f'(x) = (sec²(x) * sec(x) – tan(x) * sec(x)tan(x)) / (sec(x))².

This simplifies to:

f'(x) = (sec³(x) – tan²(x)sec(x)) / sec²(x).

To further simplify, notice that tan²(x) = sec²(x) – 1. So we can substitute this into our expression:

f'(x) = (sec³(x) – (sec²(x) – 1)sec(x)) / sec²(x).

Hence, the derivative can be simplified to:

f'(x) = 1.

In conclusion, using the quotient rule, we find that the derivative of the function f(x) = tan(x) / sec(x) is simply 1.