Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What is the derivative of tan(x)?

1. cos²x

2. sec²x

3. tan²x

4. sin²x

The correct answer is: sec²x

The derivative of the tangent function, tan(x), is sec²(x). This can be derived using the quotient rule or recognized as a standard result in calculus. To understand why this is the case, consider the definition of the tangent function: tan(x) = sin(x)/cos(x). When applying the quotient rule, which states that the derivative of a quotient of two functions is given by: \[ \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2} \] where \( u = sin(x) \) and \( v = cos(x) \), we find: 1. The derivative of sin(x) is cos(x). 2. The derivative of cos(x) is -sin(x). Substituting these derivatives back into the quotient rule gives: \[ \frac{d}{dx} \left( \tan(x) \right) = \frac{cos(x) \cdot cos(x) - sin(x) \cdot (-sin(x))}{cos^2(x)} = \frac{cos^2(x) + sin^2(x)}{cos^2(x)} \] Using the