Section 3.3: Derivative of Trigonometric Functions

  1. Differentiate the following functions. solution
    (i) \( y = 3\sin(\theta) - 4e^{\theta}\tan(\theta) \)   (ii) \( y = \sec(\theta)(2\theta - 4\cos(\theta)) \)
  2. Differentiate the following functions. solution
    (i) \( f(x) = \dfrac{2x}{x - \cot x} \)   (ii) \( y = 5t\cos t \tan t \)
  3. If \( f(x) = e^x \sec x \), find \( f'(0) \). solution
  4. Differentiate: (a) \( f(x) = x\cos x \) solution   (b) \( f(x) = \dfrac{\tan x}{x^2} \) solution
  5. If \( f(\theta) = \theta\cos\theta \), find \( f'(0) \). solution
  6. If \( f(\pi/3) = 4 \), \( f'(\pi/3) = -2 \), and \( g(x) = f(x)\sin x \), find \( g'(\pi/3) \). solution
  7. Find an equation of the tangent line to the curve \( y = x - \cos x \) at \( x = 0 \). solution
  8. Find an equation of the tangent line to the curve \( y = 2\sec(x) - 4\cos(x) \) at the point \( \left(\frac{\pi}{3}, 2\right) \). solution
  9. If \( f(x) = \sec x \), find \( f''(\pi/4) \). solution
  10. Suppose \( f(\pi/3) = 6 \) and \( f'(\pi/3) = -5 \). Let \( g(x) = f(x)\sin x \) and \( h(x) = \dfrac{\cos x}{f(x)} \). Find: (a) \( g'(\pi/3) \) (b) \( h'(\pi/3) \) solution