Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What is the derivative of cos(x)?

1. sin(x)

2. -sin(x)

3. -cos(x)

4. tan(x)

The correct answer is: -sin(x)

The derivative of cos(x) is -sin(x), which is a fundamental result in calculus. This relationship arises from the concept of the rate of change of the cosine function with respect to x. The cosine function oscillates between -1 and 1, and its slope at any point is given by the value of -sin(x) at that point. When you take the derivative of cos(x), you essentially measure how steeply the cosine curve rises or falls at that particular value of x. Since the sine function also oscillates but is phase-shifted by 90 degrees relative to cosine, the negative sign indicates that as the cosine function reaches its maximum (1), the rate of change (derivative) is 0, and as it reaches its minimum (-1), the rate of change is also 0, but in a negative direction. This change captures the behavior of the cosine graph effectively. In contrast, the other options do not reflect the true derivative of cos(x). The sine function refers to the derivative of the sine function, while -cos(x) and tan(x) are unrelated to the derivative of cos(x) and would not yield the correct result when differentiated.