AP Calculus BC Practice Quiz: Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions

Correct Answer: A

This requires the product rule, d/dx [u(x)v(x)] = u'(x)v(x) + u(x)v'(x). Let u(x) = x and v(x) = sec(x). Then u'(x) = 1 and v'(x) = sec(x)tan(x). Applying the rule gives (1)(sec(x)) + (x)(sec(x)tan(x)), which simplifies to sec(x) + x*sec(x)tan(x). [cite: 1868]

Correct Answer: A

Using the quotient rule, d/dx [u/v] = (u'v - uv') / v^2. Let u = csc(x) and v = x. Then u' = -csc(x)cot(x) and v' = 1. Plugging these into the formula gives [(-csc(x)cot(x))(x) - (csc(x))(1)] / x^2, which simplifies to (-x*csc(x)cot(x) - csc(x)) / x^2. [cite: 1869]

Correct Answer: A

This problem is best solved by first simplifying the function using a trigonometric identity. Since tan(x) = sin(x)/cos(x), the function f(x) = (sin(x)/cos(x)) * cos(x) simplifies to f(x) = sin(x). The derivative of sin(x) is cos(x). Alternatively, using the product rule without simplification leads to a more complex expression that also simplifies to cos(x). [cite: 1872]

Correct Answer: A

First, find the derivative using the product rule: dy/dx = (2x)(cot(x)) + (x^2)(-csc^2(x)). Next, evaluate this derivative at x = π/4. We know cot(π/4) = 1 and csc(π/4) = √2, so csc^2(π/4) = 2. Substituting these values gives dy/dx at x=π/4 as (2(π/4))(1) + ((π/4)^2)(-2) = π/2 - (π^2/16)(2) = π/2 - π^2/8. [cite: 1868]

Correct Answer: B

Before differentiating, simplify the function g(x) using identities. g(x) = sec(x) / tan(x) = (1/cos(x)) / (sin(x)/cos(x)) = 1/sin(x) = csc(x). Now, find the derivative of csc(x), which is -csc(x)cot(x). Applying the quotient rule to the original expression would be much more complicated but would yield the same result. [cite: 1872, 1869]

Correct Answer: D

Use the product rule, d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x). Let f(x) = cot(x) + 1 and g(x) = x - 1. Then f'(x) = -csc^2(x) and g'(x) = 1. The derivative is (-csc^2(x))(x - 1) + (cot(x) + 1)(1). This simplifies to -x*csc^2(x) + csc^2(x) + cot(x) + 1. Option D is an equivalent form: cot(x) + 1 - (x-1)csc^2(x). [cite: 1868]

Correct Answer: C

First, simplify the function f(x) using trigonometric identities. f(x) = tan(x) / sec(x) = (sin(x)/cos(x)) / (1/cos(x)) = sin(x). The derivative is f'(x) = cos(x). Evaluating the derivative at x = π gives f'(π) = cos(π) = -1. This approach avoids using the more complex quotient rule. [cite: 1872]