Ohio Assessments for Educators (OAE) Mathematics Practice Exam

Question: 1 / 400

What is the derivative of tan(x)?

cos²x

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

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tan²x

sin²x

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