Question

Can someone please tell me what the answer is to this calculus question?

Find the derivative
y = tan(x) / (1 - tan(x))

Option 1: (sec(x))^2 / (1 - tan(x))^2
Option 2: (sec(x))^2 / (1 - tan(x))
Option 3: (1 - tan(x))^2 / (sec(x))^2
Option 4: (1 - tan(x))^2 / (sec(x))

Answer 1

To find the derivative of the given function, y = tan(x) / (1 - tan(x)), we can apply the quotient rule. The correct answer is Option 3: (1 - tan(x))^2 / (sec(x))^2.

Step-by-step explanation:

To find the derivative of the function y = tan(x) / (1 - tan(x)), we can apply the quotient rule of differentiation. The quotient rule states that if we have a function u(x) / v(x), then the derivative is given by (v(x) * u'(x) - u(x) * v'(x)) / (v(x))^2.

In this case, u(x) = tan(x) and v(x) = 1 - tan(x). Let's find the derivatives of u(x) and v(x) first.

Using the chain rule, we have u'(x) = sec^2(x) and v'(x) = -sec^2(x).

Plugging these values into the quotient rule formula, we get (1 - tan(x))^2 / (sec^2(x))^2. Therefore, the correct answer is Option 3: (1 - tan(x))^2 / (sec(x))^2.