Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What is the derivative of sin(x)?

1. -cos(x)

2. cos(x)

3. sin(x)

4. tan(x)

The correct answer is: cos(x)

The derivative of sin(x) is indeed cos(x). This conclusion arises from the fundamental principles of calculus, specifically the rules for differentiating trigonometric functions. The derivative represents the rate at which a function changes as its input changes, and for the sine function, this rate of change corresponds to the cosine function. To further understand this, consider the definition of the derivative: it measures how the function's output changes as you make infinitesimal changes to its input. When the sine function is graphed, its slope at any given point corresponds to the value of the cosine function at that same point. This characteristic makes cos(x) the derivative of sin(x), indicating that as the angle increases, the value of sin(x) increases or decreases at a rate defined by cos(x). Thus, for any angle \( x \), the transformation of the sine function through differentiation yields cos(x), establishing it as the correct answer in the context of trigonometric derivatives.